Probably not what you want to hear, but don’t compete the mare until the foal is weaned around 4-6 months or longer, depending on your preference. Your mare’s #1 job is to be a mom, not a show horse. Some mares are fine to ride lightly with a foal at side, providing you have safe facilities to do so. However, I wouldn’t think of showing her (singly, not the two together) until after weaning. Just not a good idea.
That said, temporary separation between mare and foal can happen and be ok. TB mares are away from their foals for an hour or two while they make a trip to the breeding shed for live cover. The foal is left in the stall with supervision, all water buckets and hazards removed. The two should never be separated more than a few hours at that age.
I also worry when you say you “have no pasture.” Does that mean they get no turnout? Turnout is very important for a growing foal, for mental and physical development. Young horses with restricted turnout may have abnormal bone density without the gentle stress of everyday running around in the field.
I began separating my mare and foal around 8 weeks old at feeding time, as I did not want my colt sharing the mare’s grain. They can’t digest it well at that age, and it can lead to growth problems. I started by holding the colt in the barn aisle, grooming him while he ate a handful of milk pellets. Then at 3 months of age I put him in a separate stall with hay until the mare finished eating, about 30 minutes, I put them back together. As they adjusted to the routine, I left them in separate stalls longer and longer, until they were spending all day apart when he was 4 months old. By then he was basically ready to be weaned.