I'm sure that you can't do any ordinary reconfiguration of the remote to change the LCD timeout. If someone wrote an extender, you could easily change the LCD timing in the extender; Otherwise, you're stuck with the built in timing.

I doubt that these remotes have ultra low power LCDs like you find in a wristwatch. While the LCD is less power than the backlight, it is significant power. Many LCDs that size would kill the batteries in just a few days if left on continuously. Hopefully the 2116 LCD is better than that; But I still would be surprised if the batteries would last more than a few weeks of the LCD being always on (unlike watch LCDs that can be on for years).

I use rechargeable batteries in all my JP1 remotes. If I liked LCDs I'd probably agree with you and (assuming the batteries lasted a week or more) prefer to recharge batteries more often but have the LCD always on.