Well, I guess you could carry some TP and a plastic bag while you are running, it's pretty light, then take a bathroom break when the nature calls. If you are a real hero you'd pick it up and dispose on the way out. Or, just carry the TP and dig a quick hole and get it done. I see TP clumps quite often on popular trails, it's an eyesore, but when you gotta go you gotta go. I have done hundreds of hikes with a little child (my own) so I am familiar with pee and poop. When my kid was in diapers, I'd do the change, usually on the way in it would happen, then stash the diaper, and get it on the way out and throw it away at the trailhead. It's better than carrying it in the pack the entire way. But I ALWAYS got it on the way out, I never left a diaper in the wilderness.
For me, i'd just get up a little earlier and have a cup of coffee ot hot tea if needed. And to make up for it, just go to bed a little earlier.
For me I had the opposite problem, I'd drink coffee and drive to the TH, so when i arrived I had to pee like a russian racehorse. And usually there was no bathroom so I'm using a pee bottle in the car, ugh. It was a particular trail that I used to hike often, and the TH was all houses and no bathroom, so I couldn't just pee in their yard. My solution was to bring a pee bottle in the car. I was grateful to a friend who introduced me to the pee bottle many years ago, so I didn't have to go outside in the middle of the night at 18,000' and freeze my ass and other parts.