A Commission President without the support of the Council would see all her proposals shot down immediately and all her commissioners resigning. It would be untenable and she would have to resign.
There is the law and there is the spirit of the law. After all, for centuries, UK parliaments didn't even technically have to hold regular elections if they didn't feel like doing it.
I'd love to know how you've come to that conclusion with such certainty, given nothing even close to it has ever occurred, and the EU's usual way of dealing with crises is to push them under the carpet and pretend there's no problem
At least one Commissioner was forced to resign already, after being accused of collusion with organised-crime, under President Barroso. One Commission President (Romano Prodi) was somewhat marred by accusations of widespread corruption towards the latter part of his mandate (iirc generically aimed at various Commission members and nominees, rather than himself, but still there were demands for his dismissal from various governments), saw his powers effectively curtailed, and his chances of a second mandate were wiped out.
Traditionally, nothing in Bruxelles get done without widespread consensus, including staying in power.
there is no mechanism for the council to remove the commission president
the parliament can, but only by dismissing the entire commission (making it politically very difficult)
this also requires a 66.6% majority
vs the 50%+1 that was needed to appoint, so it's all but impossible (by design, again)