Bioinformatics is complicated. One needs to learn statistics, both frequentist and bayesian, since people use both; mathematics to get through the advanced probability theory, statistics and more exotic tools like Fourier series; computer science for the algorithms; programming for the implementations, usually multiple programming languages, such as C, R, python and perl; and tons of software packages, not only how to use them, but how to navigate their idiosyncrasies. Even figuring out how to get the software to run on your computer can be a chore. I find that I am constantly having to switch between Unix, OS X and Windows. For instance, a version of the software you are using might be available in one OS, but not the other. I also have to spend time learning about the local computer network (how to submit jobs to the queuing system etc). I think probably a lot of people have that issue and the rules and software change from network to network. Finally, one needs to know biology which tends to involve hours of reading. Besides this, at least in my work, it is quite helpful to know some biochemistry. I know others who need optics. So, in general, I find I have a full time job, with lots to do; and lots that I haven't done yet.
I think if you do a MSc in Bioinformatics, assuming you do well in your courses and nothing changes about the job environment, you will probably be highly in demand as a bioinformatician, because bioinformatics is in high demand, and a bioinformatician with deep understanding of the biology is even more in demand. Biologists are less in demand. I see many of the biology graduate students leaving biology, doubtful about their job prospects.
In terms of concern about doing both lab work and mathematics with equal frequency, I can think of a few subfields. I know a few people who work in optics and they seem to use the biology, mathematics and engineering all at the same time. Take a look at work in optogenetics.