>I feel like having two mortgages puts us at increased risk in case one of us were to lose a job.

That's a fair assessment, but if your numbers are good this risk becomes less and less something you need to concern yourself with.

>For context, my original plan, which was a bit Ramsey-an was to pay off both mortgages (our primary residence mortgage is a 30-year fixed at 2.25%), but I feel like that just doesn't make any sense.

You are correct. Mathematically speaking it doesn't make sense to pay down cheap debt. However, it's dependent on your risk appetite.

>I am curious about others' thoughts on either spending the next several months saving in a high-interest savings account then paying off the rental mortgage, thus freeing up around $800 a month in income, or simply investing all extra income into IRAs and taxable accounts.

I do my best to take emotion out of some decisions and let number do the talking. A little of both won't hurt either.