I've done all the permutations: taking my RHD manual and autos to mainland Europe, and renting LHD manuals and autos in mainland Europe and the US.
Driving LHD manuals is far and away the hardest. I think I naively expected the gearbox to be a mirror image, but it isn't: first is closest, fifth furthest away from you. You're sat on the other side of the same gearbox. I had some slight experience of changing gear with my right hand before I drove an LHD manual, but to further complicate matters that was mostly track cars with dog-leg first gear. It's hard work, and as I get older I've decided to not do it again.
Driving an LHD auto is easy. You're sat in the right place, next to the centre of the road, so you can see, use car park barriers and you instinctively do the right thing. You don't have to control anything with the "wrong" hand aside from putting the car in D.
Driving your own car on the other side of the road is a pain: your instincts are wrong (as you suspect) and the lack of visibility is tricky. It doesn't matter if it's manual or auto, it's still a pain.
So my advice is "rent an auto" (assuming you're happy with autos in the first place, but it sounds like you are). In an auto, you'll find it natural to be on the other side of the road, because everything's a mirror image and "just works". Think carefully at roundabouts, that's all (not a problem in most of the US, of course).