I am not a multisport athlete, so I don’t know how to square your circle, perhaps other triathletes can give you specific pointers here. But generally I’d just start very low and really flexible, perhaps 5 hours per week. Unless you are a professional triathlete, I’d make my goals for this year compatible with that. You should expect and be ok with falling performance indicators. If you manage to do better than that, count that as a win.
Tl;dr is I cut my time on the bike by 40-50 %.
Here are the details. Before my daughter was born, I wrote down my list of priorities, and for me family comes first. Then work (I have a career I love and enjoy, but is very demanding). And cycling came in third. Everything else flowed from that and it simplified my life. It was clear that I could no longer spend a month in Chile working with a colleague and good friend of mine while my wife was at home by herself. No bueno as they say here in Japan. But once I had my priorities straight, knowing what the right call is was usually quite easy. Nevertheless, sports is extremely important for me to keep my head level. (And reading your posts, it seems you are similar in that way.) I’m a very happy dad, and am glad that my daughter grows up seeing her parents being physically active. That’s an important lesson I want to teach my children.
Before my daughter was born, I‘d spend about 8-12 hours per week on the bike, sometimes more. One-day trips on the bike during one day of the weekend were the rule rather than the exception. (My wife was working weekends at the time.)
After my daughter was born, year 0, (before joining Trainer Road), I did about 5 hours per week of training plus a 4-hour group ride. But I had stopped doing those epic 6±hours in the saddle that I was used to doing before my daughter was born. Last year I joined a cycling team, bought my first power meter and a dumb direct drive trainer, joined Trainer Road and followed the mid-volume training plan. And I did my first races. I started with 277 W (3.8 W/kg) and ended up with a 308 W FTP (4.3 W/kg) at the end of last year. I stuck to the plan, except when I was sick or on business trips. (This is another thing that hasn’t been mentioned: expect to get sick a lot more.) I am thinking of upping my volume next year. I don’t know whether I’ll go to a high-volume plan or manually augment a mid-volume plan. My current FTP as tested is 311 W already (close to 4.4 W/kg), although I am optimistic the ramp test next week will confirm that I have gained a few horses.
So yes, you can gain fitness while being a dad. But you are at a much higher volume than I am, so I reckon you need more training stimulus to improve your fitness. Plus, you need to multiply that by 3.
The three biggest factors in my training have been (1) resolving scheduling conflicts, (2) sleep and (3) illnesses. Before having a kid I haven’t really been sick much. Ever since starting daycare my daughter has become the cutest incubator of germs and viruses. I have been sick about 3-4 times per year, which puts me out of commission sometimes for just a week, but on two occasions for a month. In part that was because I started training too early again (I love training, too) and the slope went from positive to negative again.