You probably need a longer cage rear derailleur. Also look at changing your chainrings. You can at least get down to 33 on a compact 110BCD chainset, possibly smaller.

Sean

If it were a shimano 10spd road setup with shimano STI levers:

A 10spd MTB cassette would fit onto a 10spd road wheel no problem ( i.e. 10spd XT stuff )
You would need a normal or shadow style 9 speed mtb rear mech to reach across a 32t mtb cassette.
You would need a shadow style 9 speed mtb rear mech to reach a 36t mtb cassette.
A 10spd rear mech has a different cable pull ratio and would not work with the 10spd STI

I have a 32t XT cassette with an XTR rear mech and 10spd Dura Ace shifters. Works very well.

I think SRAM has a different cable pull ratio, so the shimano rear mechs are out of the questions. I think the Apex range can accommodate 36t cassettes. I think Apex rear mechs have a longer cage anyhow so that may not be an issue.

Big road rides needing a 36 tooth rear cassette 😯
Where ?
What’s your chainset? If it’s a compact 50/34 then do you really think you need a 36 rear? If you’re unfit I can’t see how that can help.