Lack of power has multiple aspects to it beyond simple battery internal resistances and voltage drops.
If going with the water analogy that a lot of electrical education goes in for then your battery can provide as much power as a waterfall in some nature documentary but if you only attach a drinking straw from that into your house then you are never going to have a bath full of water. Also if you put said drinking straw next to a fat pipe you are still going to have the same problem. If going for more electrical things then this is Kirchoff's current laws meeting Ohm's law.
In this scenario the replacement screens might well gobble more than one of the small original lines could handle (one reason why , which when ganged up with by a flash cart (which might also be pushing limits, hence screen replacements and various flavours of Omega not always being a winning combo).
The black screen is one of a few possibilities. Those being the screen itself has some overheating or whatever fault that causes it after a few hours (doubtful, and if leaving the power plugged in dodges that or plugging it in for a bit but not full charge sees it drop out sooner then yeah), that the voltage drops too low to handle it (many things have a minimum voltage, the screen in this scenario reaching that before the bulk of the GBA) or that the internals that the screen is drawing extra power from overheat (hours is a long time for something like this) and shut down or otherwise malfunction.
The battery light is a basic voltage sensor that senses when something drops below a given voltage. If this was still the early 2000s and you were running on cheap AA batteries this would mean something. As this is decades later and you are running on fancy lithium that might well hit that limit lower but take longer to drop then it becomes less accurate.
Also while this is decades later then batteries have not actually got that much better in things that matter for this -- power density is still not amazingly more than other advancements would imply, most improvements coming in various aspects of battery lifetime (as in number of charges) and improving lithium chemistries to not be as annoying to handle (charging them is not a matter of putting some voltage in and leaving it like it was for some older things), store and be used by people outside a lab or high end industrial with a team to manage.
Bumping the battery up to a fancier one still might help a bit but is far from assured. Fake battery could be a thing but eh.
Options then if battery does not work, or in addition to it.
Go for a less thirsty flash cart. Not ideal but I have to note it, and might not solve the issue that much if the screen is also playing a major role.
Wait for even better replacement screens to happen -- that you have to/highly benefit if you solder something tells me that they are bodge jobs at best when someone finds something that can be twisted to work and sell it to people on sites like this for a profit. The future might not be this for a while though for while I fully expect custom batteries to be a thing (small run companies already do them and thus we are but a short skip to consumer level stuff) but truly custom screens involves quite a bit more investment and tooling than current paths in tech are likely to give just yet (not to mention low power screens are not the focus they were 10 years ago).
Figure out the internal current paths, possibly including tapping the battery in an additional place*, to provide the necessary power without stressing or pushing the currently used internal power lines for both the screen and flash cart. I am half surprised people have not done this already but I suppose the part replacement crowd is coming from interesting places rather than the usual electrical engineering inclined types with most technical focus being on software aspects.
*most taps from what I have seen are fairly low down in the circuit and picked more for suitable voltages nearby that whatever tests they did could handle it.
Or others 
