Chaosmancer
Legend
Or, you simply rule that the downed PC does not need to make death saves so long as the cleric is working on them. Say it requires concentration - if the cleric loses concentration on the spell, the spell fails and the downed PC starts making death saves again.
Using the concentration mechanic also allows the cleric to do *some* things - sling an occasional quick spell or whack someone who approaches with a weapon - while healing.
I could see that working, though it really makes life hard on the cleric (my default assumption for the healer, even though it is no longer true) to have to drop concentration spells to go and cast a traditionally non-concentration spell.
Actually, seeing a lot of people's comment reminded me of my own houserule (It's never come up because the party hasn't gone down a lot yet, and they don't have a lot of healing)
I wanted to make coming back from death feel more like a shock to the system, so I gave temporary exhaustion levels to players who were healed from zero in combat. So, for the first 1 or 2 times it happens, probably not a lot fo effect, but if you keep dropping and coming back consistently, your body starts having issues.
And by temporary, I mean they disappear once the combat is over and the PC has had a chance to fully stabilize.
Since I see no issue in coming back once or twice, and I've never seen someone come back more times than that, it seemed an appropriate rule.