And if it turns out the 6100 is dead, you can sell the adapter. Not so with a solder hack.
You could possibly get away with a VGA socket on breadboard as you say, then use test jumper clips to attach to the relevant lines on the back of the HDI-45 - non-permanent and no solder.
Apart from that, the cheapest test is going to be the AV or HPV card direct in the socket, as johnklos says, and a Mac-VGA adapter.
NB, the 6100 needs a motherboard battery to boot, IIRC. (1/2 AA 3.6V lithium, non-rechargeable) You can try the quick double-boot trick to see if you at least get a startup chime.
Check that the internal speaker is attached, and remove any socketed RAM. Switch it on, leave it powered up and attempting to boot for an hour or so, then very quickly switch it off and back on. That should leave enough charge in the PRAM circuit to boot it, all else being good. If that doesn't work, try leaving it on for a couple of hours, then overnight.
Getting a chime out of it should be your very first test.