I work as a video editor at a university, and my boss has been really gracious about covering for me when I am on-call (though in 2 years, I have not yet needed him to cover for me). I have two little guys also, so I take one client a month during the school year when I have a regular shoot schedule, and I will take 2 clients a month in the summer or over winter break. That's what I am comfortable doing. Since doula work is the employment that I like better, but it is not my bread and butter and health insurance, I prefer having one client at a time for the most part. It keeps my life in balance. I feel like I don't want to push it and end up screwing something up at my day job and losing my supervisor's support for my doula work.
If you have a job that you could do anytime, anywhere, it's not a big deal at all. If you only have a few times during the week when you MUST be at work, but you could leave in a short span of time, or you could not show up if you have advance notice and it's okay (this is my situation), it is totally workable. If you have meetings scheduled 8 a.m. to 4 p.m. every day that you cannot flake out on, I would not recommend being on call for births. Hopefully a data analyst job give you the potential to do flex-time, take work home on a weekend, or just be absent for a day here and there. If your supervisor is cool with you doing doula work, I'd say to go for it! Also, if you work for a big company, sometimes they give you 'community service time' - a set # of hours every month that you are paid for that you can use to do community service. If you work with low-income or teen moms, maybe you could even spin the doula work that way.