I don't think I can give you a lot of insight into how employers see your background, but I also had a non-traditional journey until I started working as a bioinformatician. A few things that I have found helpful were:

1. UCSD's Bioinformatics specialization on Coursera (you can also get the textbook to self-learn if you don't want to go through the course, and it's a great resource for bottom-up learning and reasoning through comp. bio algorithms);
2. independent projects that you take on to explore different techniques;
3. reading and keeping up with papers in fields of interest; and
4. staying flexible when it comes to opportunities.

You mentioned that you're working right now on diagnostics/wet-lab side. As you learn and build pipelines/workflows or implement analyses in side projects, you could try expressing to your manager your willingness and ability to do the analyses as well, if that's possible. If not, there can also be freelance, contract, or part-time positions at labs where you can gain experience as a recent grad.

Not withstanding that the job market itself is just not in a nice place at the moment, I don't think you need to do a lot more necessarily to access an entry level role.

As an employer, I would more often than not be looking at PhD-level graduates and above, but that is for roles that are not really entry level per se, where we're expecting the invidual to hit the ground running a bit more.

The beauty of coding based roles is you are likely to be judged more on your skills and portfolio than on your academics specifically. You can bolster your CV in your own time by doing small analytical projects, and building a strong Github profile.

I am also planning to pursue a master's degree in Computational Life Sciences, but I would ideally like to begin gaining relevant professional experience rather than waiting until I finish the master's.

If doing a master's, you should really find one that leans heavily into thesis or practicum-based work to garner real experience. Look into summer or year-long internships.

While I am biased, my institution (St. Jude Children's Research Hospital) provides many excellent opportunities for both internships and a new master's program in Applied Biomedical Data Sciences (all paid). Other such programs exist elsewhere throughout the US as well.

Frankly, the bar for entry is getting ever higher, and you really need to be able to point to things you've actually meaningfully contributed to - listing a vague set of skills just isn't going to cut it. The job market is rough, and you need to be able to set yourself apart to get a foothold.