The Suburban doesn't use a relay - it get's power from the instrument panel 20a stop fuse, to the brake/tcc switch, and up to the multi-function switch and out to the lights. You will need to figure out how to wire in a relay (since that is how the truck wiring was designed) or run the white wire from the brake pedal switch directly to the multi-function switch to power the stop lights (you could tie in where the truck relay used to feed the multi-function switch).
There's a splice pack from the brake switch with a leg to activate the relay on trucks, but on the SUV, it just omits this path and heads right for the switch. It's going to take some wiring work for sure, but it is doable. If I were you, I would try to pin the fuse block or use the truck fuse block if at all possible. Here's a quick and dirty diagram. Yellow is Sub, red is truck.
You should be able to make it work without TOO much grief. If you can swap the drivetrain, you got this!
*edited for formatting and words
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