1. You cannot eliminate bleed into any microphone. You cannot eliminate bleed into a piano. The resonator does it's job no matter what sounds come it's way.
2. Using monitors on stage with an acoustic piano is insane and will not allow for a puritan acoustic sound. You will have to change your monitor placement, dB level and eq curve in the monitors. FOH comes first. That is where the crowd is. Acoustic muso's using monitors is nuts.
3. It does not matter if you have SD or LD mics. Go for the tone you want. Mic'ing under the piano is a bad idea. You are mic'ing an area that is muddy sounding itself. Depending on your stage layout, it could also be the least desirable listening area. Go for the top.
4. If you are using monitors, stay away from PZMs or anything mounted on the lid or resonator. Use mics with shock mounts. I have had excellent results with DPAs, Shure 44s, 32s, 104s, etc, ATs, Sennheisers, Schoeps. Really the piano and the style of music dictated the selection and placement.
5. May I suggest visiting DPAs Microphone University on their site. They address piano mic'ing placement.

I think you mean you would appreciate a response. No one in this business appreciates "feedback"!

taxman, every time I see the word feedback on this forum I want to make that joke.

"Feedback on your mix you say? Just wire OUT to IN and press record. Mix to taste."

Sheet, to be fair a pianist may need a monitor to hear others in the setup, but I see where you're coming from.
How acoustic is it? Fully acoustic, mics only there for the purpose of recording or mics for reinforcement?

efe; Congrats on getting the gig; condolences on the troubles and tricks that go with something like this. I feel your pain, really.

As much as we wish it weren't so, the facts of modern life is that there WILL be amplified piano with some groups nowadays, including psusedo-classical and "modern" composers. If for nothing else, it's run in the monitors for other (often amplified as well) musicians.

We get this all the time at the Kimmel Center in Philadelphia; either they need to amplify it for the big hall (3000 seat Verizon Hall) or some of the instruments in the ensemble still need help, even in the smaller Perelman Theater - only 650 seats. (We just did a new work by Terry Reilly there, over the weekend: a triple concerto for Chamber Orchestra with a solo violin and 2 classical guitars. The guitars had amps, the violin did not, and neither did the orchestra.) I STILL had to mic & record it all, there's just no point in fighting it these days. You HAVE to make it work. We've had the Kronos Quartet, ETHEL, SO Percussion and many many others in there over the last five years in their "Fresh Ink' series, and all of it is amplified, including the piano. Each player usually gets their own mix, as well.

How to do it without feedback? Indeed, there are many tricks and tips.

First and foremost, it's always great to try to get the group to do it all au-naturel, if the hall will support it. Start out with nothing on at all, and see how it works. 90% of the time, sad to say, most groups just panic, and say: "Nooooo, we MUST have our amps and monitors." Too bad, but the customer is always right, eh?

My first go-to scenario, no matter the group, is usually my trusty AT-4050's in a typical large-condenser, two mic piano pickup pattern. (Go look at the Keith Jarrett KOLN concert pictures, or just about any other serious classical/Jazz 2-mic recording/amping picture. One goes over the middle of the upper part of the harp, the other points more to the back of the case, covering the lower strings. Both mics are close enough to the strings to get enough gain before feedback, but neither is so close as to get only individual notes and not a proper blend. This is tricky; and the difference is often only a inch or so between good and bad.

My second (really now the FIRST) choice is a pair of DPA 4026 microphones,with the stereo kit (4033) they sell. I recently picked up a complete system. Although a bit pricey, tHey're fantastic; best sounding mics-in-a-piano I've ever heard, bar none. Smooth and even all over the piano, plenty of range, plenty of gain. If you have the $$$, this will most likely solve everything for you.

When all else fails, DONT shy away from using PZM's if you have to; at least for the monitors. Tape 'em inside the lid and use it at full, or 1/2 stick. Yes, many folks will scream and yell about how bad they are (and yes, they generally are! Hahaha), but you have a job to do, and it can mean saving the gig (and your ass) if all you need is some good punchy piano in the monitors to keep the musico's happy. DON"T run it in the house, and don't use it in the recording, unless someone points a gun at your head. What's so bad about having four mics available on the piano anyway? Use the two good mics for the house and your recording.

Always work at getting the group to lower the volumes, regardless. Start with things lower than you'll need, or even off entirely. (It's always easier to turn it up than turn it back down, esp once they're used to it.)

You'll also have to physically set the band up to get the best position of the monitors vs. feedback, and you'll have to gently, gradually educate them all on how this works, and what yeilds the best results.

One final suggestion (which may be too late now and/or too expensive): In-ear monitors for the band will solve everything. On their most recent gig here in Philly, performing Saturn Rings, Kronos used a setup where they have two mics for each instrument: one pickup and one mic (eight total), plus theremin-like sensors in front of each player for cueing other sounds. They also have their own monitor mixers and beltpacks, with in-ears for each player. With all the cues and special effects going on, it's a life-saver for everyone, and it all sounded GREAT in the house. I had tons of good, usuable tracks for the delayed radio broadcast mix, as well; no bleed at all thanks to the in-ears.

So JoeH, how much did they fork out per player in equipment?

An older guy at our Church was complaining about the vocal monitor (which is probably more essential to us than having a spare SM57 in a recording studio)...if he knew how much a decent IEM would cost...
If I knew how much it cost...

A few thoughts
hot spot monitors move a lot less air and are easier to position in the null of a vocal mic without too much of a visual block.
Depending on the amount of boost needed in the venue, a closed lid on the piano and a couple of 57s behind the music stand has worked fine for me. I wouldn't expect that to be the solution in a very large venue, but if the speakers are providing a little boost and clarity rather than the main source of sound it might be all you need.

Thank you very much for all your posts and time.

First of all, thank you for supporting many of the ideas I had and I already tested for the group. Thank you JoeH for your extense explanation about the matter. I'm fighting to get a couple of In Ear monitors (at least for the piano and violin to start with.

So maybe I'll just keep trying and testing until I find the way. At least I have regained some confidence...

If there's still no chance of getting IEMs, you could try placing a 58 or d112 (for example; something dynamic anyway) into one of the sound holes in the resonator. Use this for monitor. usually, it'll have good gain before feedback (it'll sound despicable mostly, but...) and be of good use for monitors. Sometimes I add it in the PA as well for some ooomph :)

Your best friend is a pair of C-Ducers.  A British made contact strip you stick to the underside of the soundboard 3 “sections” away from the edge.

I have recently had excellent results with this technique. I have the lid fully open for the front part of the audience. I have a matched pair of condensers positioned at the edge of the curve of the piano mimicking that “air” for the back of the audience.

Dont forget to flip the phase of the contact mics on the underside.

PS: I like Jordan Almonds.  You don’t need to wait for Xmas. 

If you want volume, then forget tone. I suspect though, that his tour is long ago forgotten, it was 2008!