At 13 weeks a typical routine would involve a repeating pattern along the lines of 30-60 minutes awake followed by 30-45 minutes asleep. Repeated throughout the daytime.
It sounds like your DS is getting over-tired in the afternoons.
I love the bouncy chair for daytime naps, especially given you have another child to look after. Foot bouncing a bouncy chair while you are sat on sofa is a great way to get baby to sleep while still interacting with your toddler.
I like the EASY routine for this age of baby:
E - Eat - start off the repeating routine with a full feed
A - Awake time - If baby is showing over tired signs at a hour, try limiting awake time to 45 minutes or even less.
S - Sleep - Short naps of up to 45 mins are normal and absolutely fine. The key is to ensure they are frequent, which is what the EASY routine ensures.
Y - You tine while baby sleep
Then when baby wakes up you start again with E for Eat, so gave a full feed, note the time and aim to have baby back in bouncy chair and going to sleep again in 45 minutes.
The more you follow the routine, the more you get used to what works for your baby. It might be that 40 minutes awake / 20 minutes asleep works for him. Or maybe 60 minutes awake and he usually sleeps for 45 minutes - or whatever works for your baby in a very baby-led way.
The key to EASY is that while it is predictable and routine, it is done in a baby led way whereby the baby sets the timings.