You just have to know that this, too, shall pass. Sometimes I feel bad because I've been in your spot repeatedly (4 kids over 8 years plus a gestational carrier surrogacy pregnancy for a friend) over most of my career. Most working moms understand, even if they forget what it's like to have a little one. I find that more and more, working dads get it too because most of them also have wives who work FT and sometimes they have to pick up the slack at home when illnesses and other inconveniences strike. I try to be as consistent as I can and give my best when I am at work. I am careful and professional about using my time off - if I'm out of the office with a sick child or at a school event or personal appointment, I use vacation and floating holiday time to cover it. I've earned it, so I can use it. Same for you - your contract states what the deal is, so people can either accept that they may have to be more flexible with your situation or they can find a larger home daycare or center that has back up. Clearly if this were an issue, you wouldn't have clients, so they must accept it and I wouldn't worry about it!
I think that overall, my colleagues know that I give 110% when I'm there. If I'm taking time off, it's for a darn good reason and I will do whatever I can to keep my commitments. For example last Tuesday, I was running late leaving the house because I was troubleshooting a report for a client from home. I wrapped that up and dropped my youngest off at daycare, then got a call when I was almost at the office that my oldest was injured in PE and might have a concussion. So I turned around, got him from school, brought him to the pedi, brought him home, made him lunch, got him settled in and left at 1:15 PM (his sister comes home from school at 2 and they're both old enough to be home alone) to go to the office and facilitate a workshop that my boss was running that afternoon. But if it had been one of my younger kids or the injury was worse, I simply would have had to call into the workshop and send someone else in my place to assist and that would be that.
At the end of the day, I know that I provide a valuable service to my company and our clients. I get paid a fair wage, have benefits, and have earned some measure of flexibility. I do the best that I can to balance everything but that never holds for more than a few days at a time. Sometimes I work late. Sometimes I stay home with a sick kid. It all evens out, and when the kids are older I'll have years to go back to being a reliable superstar at the office. For now though, I just do my best and that has to be good enough. Be kind to yourself - I'm sure the guilt you feel is way out of proportion to any inconvenience that your clients feel. If this were a big deal to them, you would be out of business and exploring a different business model. Anything that works most of the time has to be good enough!