I thought about this while I was designing the board, but ultimately decided against it as you would end up needing a cable to go from the injector to the switch USB port (sadly the game cart socket does not break out the USB pins, thus payload injection this way will sadly never happen...) and that cable would never be able to fit into the space freed up by the removal of the battery. The cable would also have to have a non-standard end on the DragonInjector side, as there's no way you could fit the end of a USB cable inside the top of the card slot to plug it in. If you have to remember to bring that cable with you, you might as well opt for one of the larger injector solutions and just bring that.
However, the idea isn't a complete dead-end. I've been doing a lot of research on super capacitors lately, especially after learning the SX dongle uses them. We already know the game card slot has pins for +3.3v and GND. If the slot is energized when the Switch is in RCM mode and it can supply enough current then this would be an ideal excuse to swap the battery for a supercap. Technically, having no power to the game card when in RCM is still plausible, but only for jig users; with AutoRCM, a depleted DragonInjector would leave you with no way to charge it and a Switch that can't boot without a working dongle.
There's unfortunately no public datasheet for the flash chip Nintendo uses for Switch game carts, but a quick search on the manufacturer's site for high capacity flash chips (8gbit or higher) yeilds only 3 models, all of which can pull as much as 30mA. I'd be really surprised if the Switch game carts use less, especially since the chip they're using is 40% larger than the other ones I mentioned above.
Based on some quick fuzzy math, 20-30mA would be enough to charge the cap and power the trinket if I used a 3.3v 0.22F supercap. With sleep mode drawing a paltry 90uA it should last a few days out of the slot if you forget to turn the power off and charging would be under an hour. When I get some more time, I'll rig something up to test for voltage in the card slot during RCM, and if that works I'll see how much current a Switch cart pulls. Currently slimming down the board even more since, after getting my hands on a real cart, I found out the dimensions are actually 21.4mm x 31.2mm x 3.4mm.
I've been thinking about this a bit too - what to do if there's lots of interest, since the feedback so far has been really positive (thanks everyone!). I know for sure I want to release the gerbers and parts list and maybe the model too for free so that anyone who wants to can make their own. That said, while it's far too early to plan anything, I get that SMD soldering is a rare hobby. So, if this thing turns out well enough I'm open to making more to sell to the community. If I remember right, there's a license for this kind of stuff that allows me to give out the designs and stuff so you can use them for anything short of selling them yourself.