Would chrome lining a barrel after nitriding provide any benefit to barrel life?

If your ultimate goal is barrel life on a MG maybe, but on a one-off basis it wouldn't be worth it.
Not sure how well the chromium would bond to the nitrided surface, it's typical to acid etch the surface before plating.
Historically chrome lining has been less consistent than nitride, but companies like Chriterion offer chrome lined, hand-lapped match barrels.

No need.
Nitride is harder and slicker than chrome.

I had a machinist inform me that if one wished to EDM (electrical discharge machining) a part that was nitrided,  the nitrided layer would need to be ground off, since it is not electrically conductive.  I never verified that,  so I don't know if it is true.   Chrome plating is an electroplating process in which the part to be plated is the cathode (the negatively charged part) and the anode is a stainless steel rod that has lead-plated.  It would be problematic if the cathode is non-conductive.

I tend to think that chrome plating is the superior technology.

Well, I stand corrected - I just stumbled upon this report from 1968 that states that chrome-plating over a nitrided layer provides somewhat superior results.  It's on page 16 of the PDF.

Materials for Small Arms Gun Barrels