Yes install the pump. Assume you have a hot water tank and not tankless.
The pump should be on a timer and better yet would also be a smart pump (or the plug it plugs into) so you could turn it on for a couple of minutes.
The pump will return the hot water through the cold water pipe so when the pump is running for a few minutes you will not have cold water. I don't think this is a problem but it is a good reason not to leave the pump run all day.
You may want to install several hot water return values depending on the design of your plumbing. If you have a hot water line from the hot water tank to bathroom A and a different line to bathroom B and put the return value in bathroom A, bathroom B will not have instant hot water as that line does not have a return value.
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