Two of the three stems of my Zucchini plant are very healthy and have produced full fruit. One third - that on the right - is perennially unhealthy and it's fruit-buds always shrivel and fall off. I'm tempted to just chop the whole stem off, but I don't know if that will damage the rest of the plant. Any views?

I know the pot is a little small but it's all my balcony can take!

It is totally your decision. I wish that I had a magical answer for you.

On one hand, growing zucchini in a pot is very limiting, most notable in terms of available water. By removing one third of the plant, theoretically the other two thirds get more water,
On the other hand, if it ain't broke don't fix it. What if trimming backfires and that wound provides an entry point to a disease? What if it suddenly starts to produce fruit? What if the energy that it's leaves produce help the rest of the plant?
Welcome to gardening! It is totally up to you. I just think that since it is your plant it should be up to you entirely.

Its probably sickly because its stuck in a corner and not getting proper sunshine.

I feel that removing that can only benefit the rest.