Would the Game Gear have sold better (even at a higher price) if it came bundled with an AC adapter and Battery Pack?

So, as most of us know, the Game Gear was outsold several times over by the Game Boy, partially due to the phenomenon of games like Tetris, but also because it ate batteries for breakfast, lunch, and dinner to power its color backlit screen.

Now, Sega did sell an external battery pack for the Game Gear, however a better option may have been to sell the system with a (replaceable?) rechargeable battery and AC adapter, even if it increased the cost of the system. There would be some other issues, though:

Rechargeable batteries (both Ni-Cad and the NiMH batteries that became more common later in the 90’s) have a lower voltage level than similarly sized alkaline batteries. This is why rechargeable battery packs for devices that took standard batteries were larger than the battery compartment for the device-additional cells were needed, though more tolerant devices could use standard rechargeable batteries, just with less time.

Rechargeable battery packs (particularly Ni-Cad) suffered from the memory effect, where you would need to fully discharge the battery to get the most charge on the next cycle.

All that said, I think it may have been better to put that cost up front. I doubt it would have made it sell better than the Game Boy, but the system would likely have a different reputation.