The first "mod" I do to any trailer is to scrap the batwing and put on the Jack antenna.
IMO, the Jack isn't necessarily a more sensitive antenna, but, it does have a wider range of signal acceptance while the batwing is much more directional. Picture a cone extending out from the front of the antenna. The batwing's cone is much narrower and only those signals inside the cone are received. The Jack's cone of acceptance is much wider, so, a signal that was outside the cone of the batwing, now falls inside the wider cone of the Jack, giving you more tv stations and, what seems like, a better receiver. If you are camping in an area where there are only a couple of OTA stations available and they are all in the same town, then you probably won't notice any difference. If you, however, are in an area where the tv stations are in a couple of towns that are near each other, but, not in a line, then you will probably pick up more stations than a batwing would.
Winegard actually realizes this phenomenon and has come out with a new "flat plate" antenna that receives signals from a wider cone and receives them front and back equally. I thought about giving it a try, but, two things stop me. The first is that it has to be mounted closer to the roof (on a shortened mast that puts is closer to roof clutter; bad for signals, height is good, short is bad) because of my reason #2 which is that the antenna is a big, flat piece of plastic that would act like a sail in the wind. While a shorter mast may make it a bit sturdier, it is still unable to stand up to a stiff breeze and the constant swaying is, IMO, not going to do anything good to the sealant at the roof mounting... I'll keep my Jack.
I've installed several of my own and several for friends; all of them, but, one, the head assembly only and never had an issue with them not working with the standard Winegard power supply that came with the batwing antenna. If the head didn't work at all (no light on the antenna head; looks cool at night), then the head, itself was defective or the supply wire was damaged in disconnecting it from the batwing. 12v is 12v, no matter who supplies it and the system is designed to work with 12v; a small "higher" voltage won't hurt it, but, 12v is all that is required.