An established lawn is way more tolerant of suboptimal watering than new seed will be. So you should focus on giving the new grass seed what it needs.
Personally, I don't even worry about it. I overseed my cool-season lawn mid-spring and early fall, when things aren't too dry but still warm enough to encourage the seed to germinate.
But, if you want to give the new seed the best chance of germination, just focus on it. Don't over-water, just keep the lawn nice and damp. This might not even require three waterings a day, depending on how sunny and/or hot it is, because the existing grass will help shade out the seed and keep it from drying out too fast.
The advice to water an established lawn infrequently and deeply is a long-term thing, to encourage deeper root structure and (to a lesser extent) avoid moisture that would lead to fungal or other problems. Depending on the weather, if you treat the seed right, it'll germinate in a week or so, and a week or so after that you can go back to your usual longer-frequency schedule. The existing grass won't care at all — a few weeks of less-than-optimal watering won't affect its long-term health one bit — and being mixed in with the existing grass, the new grass seedlings will have plenty of shade and little risk of drying out, once they have gotten a good start on growing.