Is there a way to use Apple TV+ for free one more time?
Is there a possibility to use it somehow to launch free apple tv+ for another year?

No.
Once you have availed the free trial offer of Apple TV+, whether for an year when purchasing a new device, or 7 days for everyone else, you can not avail it again.

What will happen if I run new MacBook Pro with one of my family members Apple ID? Will it work and be visible also for me as the owner of Family Cloud?

You can not game the system to avail the free trial again.
From the Apple terms and conditions (PDF) for Apple TV+ promo:

Only one offer per Family, regardless of number of devices purchased. You can share your one-year free trial of Apple TV+ with up to five other family members via Family Sharing.