Felt recoil is very subjective and a lot depends on a shooters grip technique, skills, and strength. While weight is usually good for reducing recoil a lot of us also find certain lighter poly frame pistols pleasant to shoot. Design of the springing in the pistol including the recoil spring assembly can be a big factor as some use a double design. Recoil goes upward and straight back and the degree of each depends on many factors in the pistol design and shooters preference may favor a recoil pulse that goes more straight back while others like more upward. How well a pistol fits the shooters grip is also very important in recoil control and felt recoil. Many poly pistols come with different size back straps to help customize a pistol fit to your hand.
IMO the only real way to find a pistol that works best for a particular shooter is to try out those under consideration. I understand that can not always easily be done but it can get pretty expensive buying pistols and finding out you do not like them after the purchase.
When a shooter has fairly poor forearm strength/gripping ability, that probably is going to make it difficult to find most any 9MM pistol that they will find enjoyable to shoot.
Depending how bad the problem is, it might be a good idea to also consider .22 LR for a pistol and at least try a couple out. There are also a lot of nice ones at or close to your budget. A good quality .22 LR pistol with good quality ammo like CCI Mini Mags can be very reliable. In a self defense situation being able to hit the target in a critical zone is of great importance.
As always one should check with their doctor/health provider in such a situation to see what they say as the one thing you probably do not want to do is make things worse for yourself. I have a shoulder impingement issue and asked my doctor about pistol shooting while under therapy. He basically asked if I hurt worse than usual after a range session. I told him no, because shooting did not bother me, and he said fine go and enjoy your shooting unless pain becomes an issue.
As far as 9MM within your price range, which greatly limits options, see if you can rent a Walther Creed. I am not saying it will be gold for you but worth looking into for new in your price range. If you see used 9MM pistols that appeal to you in your price range also try to rent them at the range too. Nothing beats trying one out.
I have read many times that the Glock 42 in .380 is a very pleasant shooting pistol but that is well out of your price range.
You may also want to read this thread about discussion of felt recoil with some of the more popular pistols.
http://sigforum.com/eve/forums/a/tpc/f/430601935/m/9800050334/p/1
Good luck!