Hi.
This is the one on Continental Gateway, right by epcot, i think. If you have a handicapped sticker or card for the window, you should be able to get parking fairly close to everything. I suggest the car if you want to be able to leave whenever you want. I believe that the shuttle takes you to the Ticket and Transportation Center when it drops off and picks up, not the individual parks. So travel back to the hotel will take some time, as you will have to get the the TTC from whatever park you are at and then wait for the shuttle.
Maybe they will let you leave your trailer parked so you can take the car without it? You should call HIFS and speak to someone at the hotel (not the chain, but actual hotel in Orlando). they have been very helpful everytime i have called. 1-877-387-5437. Also get the booking at Hotel Kingdom, they have better rates than getting it thru the Hotel usually, maybe you could get the rates from Hotel Kingdom and then call to see if they will match the rate.
Also, thru Hotel Kingdom, try the Cypress Pointe or cypress pointe Grand Villas, they had great prices on 2 and 3 bedroom villas but did not have a room for our last night so we are staying at the HIFS. ( ithink they were 60$ 1 bdrom 84 2 bedrm and 100 3 bedroom per nite for the time we were going 12/11 thru 12/18)
Good luck, and don't forget to get a copy of Disneyworld for people with disabilities so you know the ropes of getting and help you need while at the parks - including special waiting areas for the entire group.