Can you get the stick and throttle quadrants to work in 4.37 setup page? Drop by tonight in TS at 20:00:00 ZULU and I might be able to help. I just tried loading the DX TM Cougar profile and it pretty much did the same thing for me, so I re-loaded my latest profile and everything was back to normal. Sounds to me like you have the most important things working which are that the BMS app is reading your HOTAS stick and throttle quadrants.
I've read on the BMS forums that some guys claimed they imported their key files from 4.36.3 into 4.37 and that worked for them which is something you could try but I would back up the entire config folder before doing so just in case it screws something up.
If that doesn't work then perhaps look at mapping everything from scratch manually which is what I do, works every time. I have my layout profile with all the DX switch mappings and I just following that, using the "BMS - Full" key file. What happens is when you click on the axis and key mapping section in the launcher, it automatically generates a new key file named "BMS - Auto" from that file, you use the alternative launcher to set and bind all your keys. You could try a hack and rename the DX TM Cougar key file to "BMS - Full", then click on the launcher and that will auto generate your new key file named "BMS - Auto" but it will be based on the Cougar profile, that's probably the quickest way to automate the process, otherwise I'd do it manually.