Electrical issue - this is new

I'm interested to know what fuse you tapped in to.
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Electronic devices not part of the car may or may not have an internal battery to maintain memory. The ones that don't will reboot or power up every time there's a power interruption. If the "V1" is a 5000 watts device or higher, I can understand the car battery light coming on due to the alternator and the battery unable to supply sufficient power in which case, the footwell fuse will blow. I don't see other reasons to cause the light to come on unless situation below.

From experience, the battery light comes on mean the car electrical system is working off battery power. This is typical when the engine is off. If the engine is running and the battery light comes on, their is a problem with the charging system and or connecting pieces leading up to the battery. If indeed there's a problem with the charging system, the battery light will stay lit constantly until the battery is completely drained and NOT intermittent. Many moons ago, I experienced an intermittent battery light issue when coming to a quick stop. It was due to the car bad fuel delivery system. The engine wasn't getting enough fuel to maintain the minimum idle speed.

A car alternator is sized to run all electronics within at idle speed. If for whatever reason the idle speed is below normal, the alternator most likely will not generate sufficient voltage. If the engine is running below idle speed, the battery light will come on indicating the car is drawing power from the battery. In my case coming to a quick stop caused the idle to briefly dip below normal and the battery light came on. Most of the time the light went away once minimum idle obtained. The problem was the dirty fuel filter limiting fuel to the engine.

Note: although the systems appear the same, my case maybe unique and therefore results may be unique. Always do your homework. Cross the T's and dot the I's.

 

I would try a direct to battery connection

You may have a bad connection that when the motor revs and vibrates the connection looses contact for a second causing the reboot.
Seen this too many times to list

Good luck