So we actually did this last summer, and yes, you can work it without having a child ride with a stranger.
1. Get FP+ for you and child #1 at a certain time.
2. Get FP+ for your friend and her child at a DIFFERENT time, slightly later.
3. Go to Mine train when it is your FP+ time with your friend both children. Tell them the 2 year old does not want to ride. (PS, the Disney policy on the website is that rider swap is for children not tall enough or the child does not want to ride. We never had a problem with that with my 3 year old who was tall enough for some rides). Get a rider swap pass. Leave Child #2 with friend and you go on mine train with Child #1 using FP+. This is the key - your friend and the two year old must go with you to get the rider swap pass. Specifically on mine train, my 2 year old and my husband were standing to the side and I have to go drag them over and present them to the guy at the FP+ entrance. I will note that DD was in a stroller, and they did not ask about her height.
4. You go back later and use the rider swap pass YOURSELF with child #2. Friend and her child go whenever it is their FP+ time.
The rider swap pass is a paper ticket, ANYONE can use it with up to 2 more people, including yourself. We did this several times last year as my husband cannot ride most rides due to extreme motion sickness, but both kids wanted to ride and I could not ride by myself with both of them.