I like the idea of interlocking the handbrake light with the remote fob, however the small bit of the circuit shown is on the earthy side of the circuit. You need to pick up the live side & the easiest way to achieve your ends is to use a relay in parallel with the warning light.
At one time modifications such as this was part of my “revenue stream” & so I’m familiar with the comings & goings of the cars wiring. The instrument cluster derived power via fuse 1 & SK1 joint & the HB lamp illuminates when the HB switch links to ground/ frame/chassis as per your picture.
I would be inclined to use a small single pole relay across these to then enable whatever else you wish to add.
M

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