I know that textbooks maintain that this phrase (even not exactly this, since it's the "incorrect" version of "we had better") should be used only in the present and future tenses but I wonder if it's a bit prudish take and now it's generally accepted to use the phrase like that.
I find this a very difficult question. Here are examples of related constructions. They use present, future and past. I never met examples of the usage "We better" in the United Kingdom, unless it is a misheard version of "we'd better".
It is raining; we had better go inside. It is raining; we would be better inside. It was raining; we were better to have gone inside. It will soon rain; we will be better inside. It may rain; we would be better inside. It had rained; we had better been inside.
No. As other comments and answers have stated, this is not possible in any variety of English that I know of.
It seems to me that “had better” functions most like the verbs “must” or “ought to". These verbs are generally considered to be “modal verbs” (related to grammatical mood). The specific phrase "had better" is not strictly classified as a modal verb by all grammarians, but in any case it shares certain similarities with the modal verbs in its grammatical behavior.
In English, certain modal verbs do not have a past-tense form, and cannot be used in the main clause of a sentence in the past tense. Instead, other words or phrases with a similar meaning have to be used.
We must go to the store./We had to go to the store.
We ought to go to the store./We should have gone to the store.
Often there are several verbs of similar meaning that can be used in an equivalent sentence about the past. The resource I've been linking to, Englishpage, recommends using "should" and "ought to" as equivalents to "had better."
Taking the example sentence you gave, "We had better work out our differences instead of breaking up," this substitution would yield the following past-tense sentences:
We should have worked out our differences instead of breaking up.
We ought to have worked out our differences instead of breaking up.
However, the two sentences above also carry the implication that you did in fact break up at some later time in the past. If you don't want this implication, I think the best phrasing is as follows:
We needed to work out our differences instead of breaking up.
Note that defective verbs without past-tense forms can still be used in dependent clauses of a sentence in the past tense:
We knew we must go on.
We knew we ought to stop.
We knew we (had) better work out our differences instead of breaking up.
It is not. However, it's possible to use the grammatically correct "we had better" in the past tense.
"No self-respecting game of make believe could start without strict parameters. “I’m a mermaid with a shiny purple bra, long green hair, and a pink tail with sparkles. My best friend is a singing octopus,” one of us would announce. If you called dibs on green hair and a pink tail, no one else had better try a similar color scheme or soon they’d be ostracized from the group and end up crying behind the banana plant."
Smoke gets in your eyes: and other lessons from the crematory by Caitlin Doughty.