I am not the radio expert, but maybe it is a compatibility issue on the wiring of that adapter?
At the top of the
instruction sheet for this product it says...
GM Data Interface with SWC 2000-2013
* For non-amplified models, or models with an analog activated amplifier
Someone will know this for sure. Does his 05 Tahoe (with Bose) have a digital or analog activated amplifier?
If digital, maybe that is your issue.
If it were me, I would get on the phone or chat with a tech expert from Crutchfield to find out if you need to order a different adapter harness to make the after market radio work with the factory Bose amp/spearkers.
This is what pops up for my 04 that is compatible with my factory Bose system.
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www.crutchfield.com
Good luck!