The solution is quite obvious: As the default pattern
[auth][year] of the BibTeX key generator could be duplicated in any other file, but JabRef have no way to check that, the ball is in your court: You must provide another pattern, ensuring that will produce always unique keys. For instance:
[auth][year][firstpage] or if this is not enough ...
[authorIni][year][firstpage]. This one use the first 5 characters of the first author’s last name, and the last name initials of the remaining authors. But if this is not enough (!) add also a
[shorttitle] or the
[keyword3]. See the help of the program. There are more options...
But if you already used
[auth][year] it could be hard check how each key have changed. And long keys are hard to remember. So another approach is construct the key based in the database, for example using
[auth][year]F for first.bib,
[auth][year]S for second.bib, and so on. Then you can predict exactly how the keys have changed. And it is still a handy key.
As far as I know, the pattern preferences are global for all databases, i.e., you cannot store a particular pattern for a particular database, but as you can re-generate all the keys in a second, whenever you want, change the preferences seem a very insignificant issue.
And well, ... another option is merge the files. You do not like that but IHMO the best option, as generate the same simple and unique
[auth][year] keys for all the references. And you can have still the source identified as keyword, for instance.
It is worth to note that biblatex can print subsets of cited references based in a keyword, so this is not a good reason to split a database.
JabRef could also work with these subsets easily, simply with a search of something as
keywords=SecondFile, so why split it?
But if you want split it anyway, once selected the references in JabRef (with this search or in any other way), is as simple as Export selected entries option of the File menu., so merge the databases should not be a problem. Just make a backup and test it!