It is mostly not about the bike or trike, it is about the experience of the riders, their power and how easy they adjust.
Without motor help a trike is usually slower than a bike, due to higher rolling resistance and the rental ones are not the race ready fast ones.
In the case of your friend it is very likely the rider as well as the cycle will be slower, making it doubly hard to keep up.
Instead of circling back, the faster rider(s) should stay with or behind the slower ones.
Rather than a sit up standard trike I would look into renting a recumbent e-trike, as those are more stable and with the motor will be able to keep up even when the person on it has little muscle power.
You will likely still allow for a slower speed, as it takes time to build up the confidence.
Better still, rent the same kind of trike for each of you and share a novel experience.
I ride recumbent trikes (non powered ones) and have friends who have e-versions. You can get them to go on their side but it is not hard to ride in such a way that it does not happen.
Go for having fun together, not for most of the group to ride their usual bikes and for one to struggle to keep up.