For me, Beemergencies get in the way of my day sometimes and don’t know how or when to tackle them when my day is busy.
Any perspectives are appreciated.
I personally love using complice.co and Beeminder together. Complice has a very deep Beeminder integration and focuses on daily tasks. Further you can add stakes to tasks.
The Complice and Beeminder combination really helped me with overload because I was able to transfer a lot of important but not essential goals to Complice and leave my most important goals in Beeminder or really any goals I actually need data on (e.g. weight loss).
Every morning, I add tasks for my Beemergencies to my to-do list and prioritize them with my other work. I had recurring tasks for this stuff before Beeminder, too, but I was not as good at doing them until there was money attached. 
I hope one day to have enough free time to add tasks for goals that aren’t in the red yet too. 
I use prioritization software that presents items on my to-do list two at a time for me to answer, “Which of these am I more likely to stay up late to do?” It’s an idiosyncratic way to work, so I didn’t bring it up at first, but when I reread this part of your post, it seemed more relevant.
If you’re in a constant fire-fighting mode like I am with lots of random work coming in throughout the day, I think ordering work based on the answer to that question might help you as well. I don’t use it for deep work, though. I block off time on my calendar for that.
If the software itself might help, I have an open source web version here, and a paid iOS version here.
Oh my goodness I LOVE that! May I mention it outside this forum, or do you prefer to keep it kind of small and semi-private?
Sharing’s fine, thanks for asking.