A 2nd battery won't help.
Sometimes the display shows up on ebay but not right now.
Only one I found:
NEW LCD Display Screen For Pentax K S1 Digital Camera Repair Part|Camera Modules| - AliExpress
There are more on aliexpress but more expensive as well.
Most important part for disassembling the K-S1 here:
Manual: Solenoid replacement: Pentax K-S1 - PentaxForums.com
when you are that far, you'll find out how to remove the back-part with the display as well.
I haven't done it with the K-S1 but with a few K30/50 and K5/3 as well as K100D/K200D.
Usually the back-part with the LCD comes off by disconnecting just 1 or 2 flat ribbon wires from the mainboard.
Then take photos of each step.
Could be that the display is alright and its just the connector on the mainboard!
As you have your K-S1 open anyway, I would exchange the solenoid as well then!
50% of all K-S1's we have in the family and with friends developed the dark-image-syndrom (but all were pre-12/2015!)
You can check here the manufacture-date:
Check Camera Shutter Count and Manufacture Date
Only if post-12/2015 I'd risk it not to swap it.
Don't even consider the stupid filing/sanding job, somehow the K-S1 goes quicker mad when this bad solution is applied!