Should I buy a Dell Latitude 5500 over a Latitude 5400?

The only real difference is the screen size -- 14" or 15".  Whichever size you want, purchase that.  There's little else to differentiate the two systems other than size and weight.

@ejn63 ​ I am aware about the screen size differences but I’ve heard the Latitude 5500 has insufficient fan control and has trouble with sudden heavy work loads. Is this still true now?

That would be a question for whoever made that claim.

"Heavy work loads" is a very nebulous term -- you're asking about a system that's almost five years old, so it's not going to perform as a brand-new system would, if that's your metric.  You haven't specified the CPU, but that plays a role.  And if the "heavy work load" involves heavy use of the GPU, then neither system is going to be a spectacular performer.

All else equal, a 15" system should run cooler than a 14" system, but note that the condition of a system that's almost five years old plays a role.  Just as a well-maintained used car from a manufacturer with a track record of reliability can be a good buy, a poorly maintained used car -- from the same manufacturer -- can be a risky buy.

Neither system is designed for use in gaming (or any other video-intensive work), and neither are they designed to run heavy calculation loads for hours on end.

@ejn63 ​ I heard that from Notebook Check, here is the link to their review of the Latitude 5500.

https://www.notebookcheck.net/Dell-Latitude-5500-Review-A-business-laptop-with-many-options.434050.0.html#toc-6

That review is the same age as the system -- so subsequent BIOS updates that may have changed the thermal profile of the system aren't considered.

@ejn63 ​ In the review?

The review was made in 2019 when the system was new -- based on the system as it existed back then.  It isn't unlikely the thermal profile may have changed since then as newer firmware was made available.

If you're looking for a system capable of "heavy workloads", and that's paramount -- particularly if the system is for business use, you'll do better with a new system having a warranty.  This system is almost five years old -- bear in mind the design life of a notebook computer is three years, so consider downtime if you need to rely on the system.  Dell considers anything older than four years as end of life, so whatever service you may need will be by you or paid for thorough a third party.

what about Latitude 5590?