TBH, I think that getting it out of the cupboard (even if it’s temporarily for testing purposes) is a good first step.
I’m not familiar with the specific GFX card in question, but are you sure that the exhaust is the exhaust? The dual-height cards I’ve seen vent hot hair out of a duct at rear of the card, the bit that you’re blocking is an intake.
Either way; is this HSF assembly on the GPU clear, or is it full of dust?
Do either of the PCI boards generate heat? Is there any value to swapping over the two cards? (I’m thinking the TV turner might get a bit warm…)