Does anyone see any problem with this offer?
Yes - I see lots of potential problems.
I assume you gave an honest answer when asked how long it would take. Now you are being asked to complete it in 60% of your estimate.
So were you lying earlier? Or are you planning to cut corners now?
Either way, you might inadvertently be setting yourself up for future conditions where management doesn't trust your estimation ability ("He said 5 weeks last time, and got it done in 3. Why should we believe his estimate of 5 weeks for this new project? We know he could do it in less if he really wanted to!") or believe you deliver shoddy work ("We gave him vacation money, but his project is full of bugs!"). Your actions in this case might set a precedent.
Or is it just comparable to a cash bonus for completing a project?
It's more like a "cash bonus with major strings attached" to me.
Should I continue to stand my ground with the realistic deadline?
Clearly that's a decision you need to make on your own, based on your own beliefs on how to do your work well.
Has anyone faced a situation like this before and what happened?
I've been involved in situations where I was told that my estimate was unacceptable, and that I had to find a way to get it done in less.
In every case, I've discussed the ramifications ahead of time. I've explained what I saw as the tradeoffs between time, money, and quality. And in each case, I've tried to make it explicit what the chosen path would involve - both positive and negative.
Sometimes it worked out well. In those cases, management was will to consciously make the tradeoffs I was offering.
In a few cases it didn't work out so well. Management apparently believed they could get "faster" without trading off anything, even though I told them ahead of time that it wouldn't work that way, and they were unhappy with the results. Perhaps I wasn't convincing enough, or perhaps they only really heard what they wanted to hear - I couldn't really tell.
I'm in QA. These days, my motto has been "We can test anything, in any amount of time. But we can't do it as well. Let's talk about what that means...".
I always try to understand the scheduling dynamics before committing to a deadline, and I'm always as open and transparent about that as I can be. Schedules and deadlines are a business decision. Management can decide whatever they like and I'll do my best to make it happen. But I insist on being honest about my assessments and estimates.
The fact that you used the word "bribe" in the title, and that you expressed specific concerns about quality, tell me that you are conflicted here - and rightly so. If I were you, I'd talk with your bosses more. Try to explain how you think quality will suffer in as specific a sense as you can. Then see if they consider that acceptable or not. Let their reactions be your guide.
Sometimes deadlines are truly important, "either we make this deadline or we close up shop" kinds of deals. Sometimes important events (shows, customer meetings, holidays) help drive deadlines. But often, deadlines are just made-up dates that have little real meaning. Try to understand if and why 3 weeks is so important in this context. Sometimes that can help make your decision easier.