Almost every single incompatible app I've loaded onto the Kindle Fire works perfectly. Maybe one exception in 20 different apps.
Here are two ways to get your incompatible apps to your Kindle Fire using your Android phone, when obviously they don't appear in your Kindle Fire's Play Store:
Method 1:
1) Install the desired Kindle app on your phone first.
2) Use Titanium backup to backup the newly installed app on your phone.
3) Within Titanium backup on your phone, click the new app's backup and use "Send to..." to email the app to your Kindle Fire's email acct , or your PC, or wherever.
4) Get the .apk to your Kindle via USB or whatever, (if you didn't directly email it there).
5) Install the app on your Kindle (click from within ES file explorer, apk installer, etc.)
6) Betcha it works on your Kindle!
Method 2:
1) Make a Nandroid backup with Rom Manager or directly from CWM on your phone.
2) Install the AppExtractor app on your phone.
3) Extract any/all your apks onto your phone SD card via the phone's AppExtractor application. This extracts apps from your Nandroid backup. You can select from which backup.
4) Copy the desired apk from your phone's SD card, where AppExtractor stored them, to your Kindle, via USB, or email it, or whatever.
5) Install the app on your Kindle (click from within ES file explorer, apk installer, etc.)
6) Betcha it works on your Kindle!
A few notes: Operable, yet technically incompatible apps moved from your phone to your Kindle will show as installed apps in the "Play Store," but it won't let you update them. I've seen very few apps that won't work on your Kindle Fire, and those that don't will force close or have strange graphics. Some, like SpeedTest, work and look fine, but are "phone sized" on the Kindle display. I haven't seen an app that screws up the Kindle, though some misbehave as previously mentioned. Not to say you can't cause harm... your mileage may vary. (<--- disclaimer)