Air carrier pilot reported bleed system overtemp in cruise in icing conditions. Flight crew diverted and landed safely.

Narrative

After leveling off at our field cruising altitude on the way to ZZZ we began experiencing light to moderate turbulence. Traffic ahead of us had climbed to FL280 and reported a smoother ride. We had numbers for FL280 so we requested it and were given a climb and maintain FL280. At around 24500; we began picking up icing and noticed the anti-ice system responded correctly with four open" indications on our ice panel. Moments later at around 25000 ft. we received a bleed 1 overtemp message on our EICAS. I ran the appropriate QRH while the first officer continued flying. The QRH instructed us to fly a maximum of FL25000. The QRH also advised that if we were to pick up icing with one bleed inoperative; we would have a maximum altitude of 15000. At this point we were around FL260 and not in icing conditions so we notified ATC and asked to descend and level at 25000. We then notified Dispatch. We were in a thin cloud layer so we descended to 22000 in order to find warmer air temperature and increase the chances of not picking up icing. Updated ATIS at ZZZ advertised an east landing operation which prolonged our route (we had planned landing west). Combined with the lower cruising altitude we realized we would be landing at around 1900 pounds of fuel and if we picked up icing we would have to descend to 15000 ft. making us even tighter on fuel. We notified Dispatch again and asked for long range cruise numbers and an alternate of ZZZ1. We coordinated with ATC and slowed our speed on arrival. We landed with about 2250 pounds of fuel. We notified maintenance after parking and wrote up the bleed 1 overtemp.Suggestions: I believe we handled this well as a crew. Checklists were completed in a timely manner and we made sure to keep dispatch and ATC advised. We also double checked the QRH and made sure we had done everything correctly. We potentially could have asked for an alternate earlier in the process."