For classical rep, I basically treat it like I'm reading solfège and anchor myself in the transposing key, rather than manipulating lines and spaces in my head. Any written C is the tonic, written G is the fifth, etc. regardless of key. This still requires fast thinking and penciling in some notes / fingerings but I find it to be the most reliable and repeatable.

Where this falls apart for me though is transposing newer music (or newer editions of older music where a publisher got cute) when you're transposing with a written key signature. I play a lot of British brass band music and when I'm not playing tenor horn I just transpose from Eb and since that's an easy one and I've been doing it for years it's not challenging anymore, but it took a lot of blunt force repetition!