I doubt it's the battery. Never had a problem with the same DVD batteries powering an R44 and/or a V3 in the same situation other years. I've only paired the V3 with the 680 in less thermally changed conditions, but when I do it's arranged right under the recorder, seperated only by the optional snug fitting cover my 680 lives in, both of them inside the main compartment of a soft-sided cooler bag (itself insulated, if weakly) along with the batteries.
The time I had the issue was in very hot direct sun, cooler bag zipped up to protect from ramdom beer spillage except a small opening for cables. Recorder was running constantly for a few sets while I was away. Came back to check things and start a new file between sets and found it very hot. It was over a year ago and I'm sketchy on what the exact issue was- may have been oddly pulsing levels, menu freezing, or somthing like that. I powered down let it cool a bit, powered back up then proped open the bag lid and kept it shadded with a Gakbrella or white rain towel. Worked fine the remainder of the weekend and ever since. I've kept the bag shaded and lid cracked in similar hot conditions at the same place since and the problem hasn't re-occured.
[edit- I was running phantom on all 6 channels at the time - significant, but probably not an overly huge current draw. I'd have to go back and check to be sure of which mics, but I think it was 4 AudioTechnica PFAs powering the Tetramic + 2 Niant PFAs powering 4060s. At the same hot situation place since, I've also run 4 Niant PFAs w/4060s + a pair of Gefell M94/M692.