“Let's burn that bridge when we come to it” – is this sort of idiom mixing considered a pun, and if so, does it have a specific name?

Seems these are called malaphors

Definition:

An informal term for a blend of two aphorisms, idioms, or clichés (such as "That's the way the cookie bounces").

Etymology:

A blend of malaproprism and metaphor, coined by Lawrence Harrison in the Washington Post article "Searching for Malaphors" (Aug. 6, 1976)

Wiktionary has your exact example

Examples include "hitting the nail on the thumb", "barking up the wrong alley" and "We'll burn that bridge when we come to it".

I believe the term for this is mixed metaphor

This is where you take two common metaphors and mix them together, often incorrectly, to make a metaphor that doesn't make sense.

For example, mixing "You can't have your cake and eat it" with "It's not over till the fat lady sings" might produce "It's not over till you've had your fat lady and eaten it".

If your particular example is meant as a joke, then I would say that it is a pun.

It's fun to use a mixed or reverse metaphor deliberately and which elicits corrections from jerks who don't get the joke. Three of my favorites are: 'If you can't laugh at other people, who can you laugh at?'; 'It's as plain as the face on your nose!'; 'I thank you from the heart of my bottom.'

We have been calling these mixaphors here; a portmanteau of mixed and metaphor. We think they are the funniest thing since sliced bread.