if you were to insist on the active form of "to bring along" I too would say
Die Frau, von der er darauf bestand, sie mitzubringen? is correct, I feel it's not really how we would commonly say it though, at least not in connection with insist / darauf bestehen.
My feeling is that the passive would be used much more frequently and it makes the sentence a lot more straight forward
"Die Frau, auf deren Mitnahme/Beisein/Anwesenheit er Bestand," war deutlich angetrunken ...
I think the main issue is the example you chose as there is no passive for the verb mitbringen, which makes me feel that in this case it should be replaced with Mitnehmen / Mitnahme or Beisein (presence), especially as you seem more interested in how to generally translate those type of sentences.
If there was a passive for mitbringen the sentence would most easily be translated as
"Die Frau auf deren """Mitbringung""" er bestand" (though there is no such word).
other examples are:
the mountain which he insisted on climbing:
active: Der Berg, von dem er darauf Bestand Ihn zu besteigen ( was 3000 meters high)
more commonly though: Der Berg, auf dessen Besteigung er bestand...
the car which he insisted on buying was ...
Das Auto, von dem er darauf bestand es zu kaufen...
more commonly though: Das Auto, auf dessen Kauf er bestand...
as a rule of thumb I'd say once "insisted / darauf bestehen is involved the easiest and most common way to translate these sentences is by switching to the passive. Unfortunately the example you gave (bring along / mitbringen) has no passive, so we would usually have used another word to begin with (mitnehmen/ Mitnahme) which would mean no such construct as "Die Frau, von der er darauf bestand, sie mitzubringen" would usually be used in every day language.
In most other cases (suggest or recommended or intended etc. instead of insist) both versions would work in a rather straight forward way:
The Woman he recommended to bring along was a scientist:
die Frau, die mitzubringen er empfahl war Wissenschaflerin
Die Frau deren Mitnahme/Beisein er empfahl (passive , but there is no passive for mitbringen so changed to mitnehmen again)
The Mountain he suggested (vorschlagen) to climb was .. :
Den Berg den zu besteigen er vorschlug war.. (active and not uncommon)
Der Berg, dessen Besteigung er vorschlug war... (passive)
The House he suggested to buy
Das Haus, dass zu kaufen er vorschlug... (active)
Das Haus, das er vorschlug zu kaufen ...(active)
Das Haus, dessen Kauf er vorschlug... (passive)
Maybe the brainiacs out there could list some sort of rule? If not I'd say with insist use the passive of the verb, if there is none use another word instead, though grammatically what you suggested is 100% correct (just not a native speakers first choice).