I don't think it would hurt an established tree like that very much at all. It might make some young growth immediately after pruning that might get damaged IF we have a frosty winter, but that would be just a minor issue really. For young trees we wouldn't want to do heavy pruning that would cause a ton of new growth that is vulnerable to winter damage, but for a more established tree I wouldn't worry about it. Keep an eye on NOT suddenly exposing branches/bark/trunk to sun that were totally shaded by foliage though. That could cause a problem if October is sunny/warm. If it's just some minor/medium trimming I wouldn't worry. Let's see what others say too. Let us know what you do and how it works out!

I would be very wary of trimming Jacarandas at any time. They have a habit of developing unsightly fast-growing vertical branches wherever they are trimmed. I understand you have to accommodate your neighbor, but recommend keeping it to a bare minimum.

At this time of the year I would trim minimally to make your neighbor happy. Trimming now will encourage new growth that will easily freeze in our lower winter temps. I would wait until late winter early spring to complete the major trimming. it will be easier to trim then as the tree will have defoliated and you will be able to clearly see all deadwood and crossing branches :). Deana Bella Fiore landscape Design