A current ME/IR is only needed for issue of your first MPA type rating.
Your 737 type rating gives you type specific Multi Pilot Aircraft/ IR, therefore the "MPA/IR" is revalidated with the LPC of the second type rating. So yes you can commence another MPA type rating with the previous one not current.
However while regulatory wise you do not need your MPA/IR to be current there are several employers that require a current IR as a company / recruitment policy. So if you are self funding a type rating it won't be an issue.