Since no one has come back to you I'll just give a couple of points.

Firstly, none of the vines on your pergola is actually Ivy. I would suggest you try asking over on the Vines Forum or the Name That Plant Forum for correct ids for all three vines. People over there will need to know your location and see some detailed shots of the plants. The blue berried one looks like Ampelopsis cordata but I am not familiar with it in person.

Your vines will be unfazed by being cut right back. The problem will not be whether the Trumpet vine grows back but whether the other two can be killed off. It's not a problem where I live but a lot of people in the US consider Campsis radicans (if that is the Trumpet Vine you have) to be a nuisance in a domestic situation.

Clarification on the term "ivy" which refers primarily to plants in the genus Hedera. This LINK details the various species of this genus.

Indoor ivy species are often named for their leaf resemblance to Hedera. For instance, Swedish Ivy is in the genus Plectranthus, but is native to tropical Australia and various Pacific islands.

More than just a nuisance, Campsis radicans can grow under siding and roofing to invade the actual house. If not controlled carefully, it can literally tear a house apart.

Martha

I do not see Campsis radicans in those photos