He'd need a training course first? That seems really odd to me. Maybe he's just uncomfortable with it for whatever reason and is using that as a sort of excuse?

I haven't done a formal walk and talk, but with ex-marriage counselor, a couple times when I was having a panic attack in session, we walked around the outdoor courtyard of the building for about 15 minutes, continuing talking about therapy stuff. I did that once with ex-T as well. Maybe that was a little different because it was an internal courtyard, but there are other (nontherapist) offices in the building, and some had their windows open, so we could have been overheard.

It wasn't walking, but I've also met with my current T at an outdoor table at the coffee shop downstairs from his office a few times (due to the pandemic), so we were essentially in public. Actually, I guess H and I met inside that coffee shop once with ex-MC, and he said he often met clients there.

I'm wondering for your T if it's more about being outside of the office, the actual walking (like, I don't know, what if you trip and injure yourself, though I imagine you could sign a waiver), or some combination of them? Though I'm thinking from a US perspective, so maybe it is different in the UK? I hope you'll be able to work out a way to walk with him at some point.