Can you request an MOS change?

Easy to change if you re-enlist, but to do it otherwise, and others have said is rough. The Army paid to train you to do one job, and they won't be thrilled to spend the money to train you again without getting more time. My useless two cents from an old fart, but I did work in re-enlistment for a few years and did a number of contracts for people to change.

It sounds like they want to send you back through selection again. If I remember correctly, you should be MOSQ as a 31B already. There are no slots for a 31B at a PYSOPS battalion, or any other support battalion

You can switch to 19D pretty easily in the somewhat near future. Your MOS has to be Overstrength and your requested MOS has to be under strength for you to request a voluntary reclass. You also have to have at least one year on station prior to your class start date. 19D is frequently under strength and 31B is frequently over strength. Another option is to pass Ranger School and to reclass to 11B, there are plenty of recon opportunities in the Infantry world.

Due to the injury, are they going to recycle you? If not, then your reclass will probably be at the needs of the Army

Would this injury prevent you from performing as a 19D?

You can request it, but you may be out on a waiting list until a slot with your rank comes open. It's probably best to wait until you got E4 it's the level where shit gets done and you can never have enough E4s. Besides everyone knows about the E4 mafia!!

S.M. .. if you are still active duty even tho you are in a recovery you could possibly be re classified as the Army or Service sees fit .. the service trained you kind of quick and quietly to help you but to help themselves as well . there is protocol that must be followed and do that to the best of your ability and get help from H.R. (human resources) or your S-1 part of your unit .. however if no response go up the chain of command legally and properly. do your best .. stand up, stand out, and stand above and show that you know and you will go .. go places to help you be a very great soldier ..

SFC (Join to see) replied with expertise as a Career Counselor. I'm FAR from being a Career Counselor but will add my 2-cents as I requested to reclass from a 91B (68W today) to 95B (31B today). I was told at all levels that it was EXTREMELY unlikely such an MOS change would occur because Medics had a VRB (Variable Reenlistment Bonus) of 4 ($10K) compared to MP with a VRB of 1 ($2K). Both MOS were critically short; but the medical MOS was considered a more urgent need.
I stuck to the medical for six months before submitting my paperwork and during that time I accepted EVERY duty detail that came up. In short, I busted my butt to impress my Section NCOIC, the Helth Clinics Emergency Nurses and Doctors, and my 1SG. In other words, I tried to do every job, no matter how minsicule or how detailed to the utmost of my ability. Along the way, I happened to impress a LTC during a medical emergency and when he asked what my goals were, I told him I was trying to reclass to the MP Corps. As it happened, he put in a good owrd for me and that tipped everything in my favor.
Moral of the story? Don't let anyone tell you that youo "CAN't" reclassify - assuming that you meet all criteria for whatever MOS your going for - physically, mentally and security classification, etc.

If you slated for recycle there is a 100% chance your gonna be recycled

Of course you can. Whether or not that request will be considered or honored is a different story. Usually, you'll receive "no" or "F--- NO" from your chain of command. So be all the way squared away before you approach them with your request, have a convincing enough proposal or argument as to why the reclass will be the best decision all around, and be prepared to be told "no" in some way, shape, or form.

One of the best kept secrets in the US Army is that you can REQUEST ANYTHING -- the chances of approval and the level at which a request can be approved or denied will vary but you can request absolutely anything.