It's come up before on this site, see mathoverflow.net/questions/431499/…

There's a pullback (restriction) map on differential forms $\Omega^j_X \to \Omega^j_Z$, and it sends compactly-supported forms to compactly-supported forms. Now just define the map $D^i_Z \to D^{i+r}_X$ to be the dual of this. No need to even mention $\delta_Z$.

Just a suggestion: have you tried considering maps $D^i_Z \to D^{i+r}_X$, rather than $\Omega^i_Z \to \Omega^{i+r}_X$?

(sorry, I posted a comment but it was nonsense). This looks great, thanks!

@WillJagy I've added a clarification of exactly what I'm answering (since the title of the question, the first paragraph of the question, and the final paragraph are actually asking three distinct questions)

I'd be very surprised if this were implemented, it's already pretty difficult to compute the action of Atkin--Lehner operators in integer weight (there are no simple formulas for the action in terms of q-expansions).

Incidentally, I was working in p-adic automorphic forms / eigenvarieties roughly between 2005 and 2012. During this time, I read many, many papers, and wrote a fair number myself, in which Tate-style rigid spaces appeared. I can count on one hand the number of times I encountered Berkovich spaces during this period. (Do not fall into the trap of judging what was "generally known" at some point in the past from a few landmark papers of that time that are widely read nowadays; these are, by definition, unrepresentative of the time they were written.)

I disagree with your claim that "Berkovich spaces were what was supplanted by adic spaces after Scholze popularized them, rigid spaces as originally defined had already fallen out of favor for many". During the decade or so prior to Scholze's arrival on the scene, Berkovich spaces were used in some areas as an alternative to classical rigid spaces, but they never came near supplanting classical rigid spaces across the board, as adic spaces later did.