My gut is the regular Hilton LBV simply because I think I've read it's a closer walk to Disney Springs and that's important to us. The pool at the Palace sure looks nice though! Lazy river! Hello
Unless you have mobility issues, the difference in distances from either Hilton to Disney Springs should not enter into the equation, as far as I'm concerned. In fact, I'd say some rooms in the Island Building at the Hilton Buena Vista Palace (BVP) are probably closer than some rooms at the Hilton Lake Buena Vista (LBV). Even if you're in the Main Tower at the BVP, you'd only looking at a few extra minutes of walking.
I've stayed many nights (40+ in the last few years) at the Hilton LBV as a Honors Gold member. I never had a perfect stay. Major issues included a broken toilet that took 3 calls and over an hour to get fixed, Housekeeping ignoring "Do Not Disturb" signs on multiple occasions, and being woken up by groundskeepers using leaf blowers before 7am. The staff response to these incidents was not ideal.
For me, I'll only stay at the Hilton LBV if the price is below $110, including the resort fee. I'd never consider paying the 50,000 points/night that they want unless it was an extremely busy time like a holiday or Spring Break.
As for the Hilton BVP, I've only had one experience. I checked out early after one night. Honestly, the room was the dirtiest hotel room I've seen in over two decades. That experience was at the start of their renovation (where they added the new pool). I haven't been back since that happened. If they did a deep cleaning of the rooms with that renovation, that's probably the better choice.
Finally, I'll talk up the DoubleTree Suites. I've easily stayed there 60+ nights in the past couple of years. I've literally never had a major problem. Any minor issues (toiletries not replaced, a broken in-room safe) were fixed with a single phone call. The staff is above average. I had a minor dental emergency once while there and the Front Desk was fantastic. Every room is a two-room suite. They did a room renovation a few years ago.
The major downside is that it is much further from Disney Springs. So, I'd understand why you might not be considering that property.