Pumpkin82... You say you used to get on OK with your MIL before baby arrived, but now you admit to being a little less intolerant. You say your MIL says things without consideration to how they land ... has this always been her way, or only since you became less intolerant? And, because of this you say you no longer want to pop in when passing, or to arrange a catch-up, so you've stopped doing this leaving it up to your DH who isn't very proactive with making arrangements. I would think that if prior to baby arriving you occasionally popped in, and now no longer bother to do so, that your in-laws probably feel that you no longer care about having a relationship with them.
Your baby was a covid baby, and we have all seen many posts on GN from DILs complaining about GPs still wanting to visit their GC, when DIL doesn't think they should do so. This has caused upset, as the GPs have been seen to be selfish, and irresponsible. You say that your in-laws have not seen much of your baby since he was born, but do you think that possibly they may have been concerned GPs and not wanted to take unnecessary risks due to covid? Sometimes it seems that whatever GPs do, they are often in the wrong.
You say they aren't very communicative, but it sounds like neither is your DH, and you've admitted to also not being bothered anymore to contact them. If I'm honest, I never called my MIL either, because I always felt a bit awkward about talking to her on the phone, but we always made a point of visiting once a week with the GCs, and they had a great relationship whilst growing up.
Regarding the birthday present on the doorstep incident. The fact that it was dropped off on the doorstep, and then a text was sent to MIL to let her know it was there, I would find extremely upsetting, and quite frankly I'm not in the least bit surprised that she was miffed, and has brought it up since. Would a phone call have been so difficult to say happy birthday? Or perhaps you and DH were just a bit embarrassed at the fact that you left the gift on the doorstep? Would you have allowed that to happen were it your own mother? We all know that babies have wants and needs, but taking baby out of the car for 5 minutes to pass on the gift, and wish MIL a happy birthday wouldn't have done any harm. He may have simply been objecting to being strapped into his car seat, or simply needed a nappy change, or a feed ... both of which I assume you'd have taken along supplies on your car journey, and would easily have been able to be tended to in GMs home. I would say that your MIL had something to be upset, and 'cheesed off' about.
You say that you feel your in-laws aren't remotely interested in you, but perhaps under the circumstances they feel the same about you and your DH, as neither of you any longer bother to pop in to see them, or to call them.
"Has anybody got any suggestions or thoughts as to how to handle this? We haven’t seen them for a while and DH seems to think they won’t be very impressed if we don’t see them soon because they will expect to see us."
My suggestion would be, if you do want any kind of relationship with your in-laws, it's not so difficult to be nice, and simply invite them round for Sunday lunch or somethings. Insist that DH does it if you don't want to call them. If they choose not to come, then you know where you stand with them.
I know that there are other posters here who will not agree with my views, but I simply wanted to point out that having read your OP, different people interpret things in different ways. Some with say your in-laws don't deserve to be included in your family life, because they don't communicate with you, and don't seem to care. Others will look at it from a totally different perspective, and feel that they may be a little upset, and deserve another chance at being a part of your family life, so that they can enjoy being GPs, and your little son can enjoy having them in his life.