No they are not prone to heat bumps. That is an "old husbands tale."
They do seem to be prone to allergies.
It sounds like he has hives and they are flaring up periodically.
Have you changed his food?
Any new treats or people food?
Has anything changed in his environment?
Ah-ha. I bet when he was outside with the puppy he came in contact with something he wa allergic to or was stung by something. The hives can keep coming back from that one time.
Well I think we can fix this.
How much does he weigh?
Give him 50 mg of benadryl by mouth, every 8 hours for the hives. Once the hives are gone, keep doing the benadryl another 2 days just to make sure they are not going to come back.
Once he is off the benadryl, if the hives do come back, he may be allergic to something in your yard or house. There are allergy tests that can be done at your vets if you are interested.
Or, you can treat him with benadryl when he gets hives.