I understand the urge to improve the beloved Box, but don't get excited about changing battery types from NiMH to Li-ion. It's not happening here. The rules are against it. To do so you'd have to radically change the screen and trough.
MFLB is based on the idea there's about .75 Volts 'across the rails'. Of the original 1.25 or so the battery has half a Volt (give or take) is lost to internal resistance due to the 'out of spec' 15 Amps we demand. The original cell used was never designed for that level, when pressed the terminal voltage 'sags' just like the kitchen lights dim when the washer starts. The abnormal current demand of the starting motor causes 'voltage drops' through all the wiring in the house, circuit breakers, meter, feeder to the house, and power lines up to the local transformer up on some pole. In fact, in normal operation houses closer to the transformer 'see' higher than those further away. The power company sets up around what's called a "load center", some ideal customer mid way along the lines with everyone along the line getting more or less voltage than ideal. A compromise.
This, voltage loss is why the battery on MFLB gets hot as well as the screen. It heats about 2/3 as much as the screen does (in Watts) since 2/5 of the voltage is dropped there. That half a volt times the nominal 15 Amps means we're 'wasting' about 7.5 Watts delivering 11.25 Watts (give or take) to the screen welded between the rails that forms the trench.
Enter the humble box which now expects that .75 Volts (down from 1.25 nominal) but will now get much closer to 3.6 on average, and as high as 4 plus on full charge. Very little drop since the current levels are much less due to the higher voltage.
The size of the wire that makes up the trench, and it's length, determines that current (and therefore the power) as Ohm's Law specifies. To work correctly with the higher voltage we'd have to make each strand 3.6 over .75 times (four) times longer. A much bigger trench which is now weaker because of that extra unsupported width. The trench, which now has trouble dealing with our poking and prodding in use, would be all that much less able to cope.
Or we could make the cross section of the SS wires 1/4 what it is now, and therefore weaker for that way.
MF, through it's development, is 'married' to the battery type it uses. You can't simply go to a new type with more capacity, attractive as that is otherwise, without dealing with that very serious problem. Like adding electronics to convert the voltage to what the trench needs to survive.
No way around Ohm's Law........which is why it's a Law.
Nice dream, but reality is against it. Sorry. MF really 'knew their business' in this as I understand such things. That 'inferior' battery is really a top pick, the only one possible with the basic design. Li-ions simply won't work. They would combust the load immediately, get 'glowing hot' and most likely fail on first use.
OF