My experience is that the FileMaker office is very helpful in supplying links to older versions. I think, however, that the licence number would be different, however this generally appears with the link that they send. So no, I don't think it's a good idea to use the v16 licence key with v15. (but I might be wrong).
Call FM Support - You can buy a current version of FileMaker and get a 'downgrade'. You can't use both versions - but You get a FileMaker Pro V15 (at least, one could do this in the past. I'm not sure about single versions)
Besides of this, FM16.02 works quite well - with V15 databases (it will not connect to FileMaker hosts older than 14.04 (FMS), needs the latest V14 patches)
I may just end up upgrading to FM16 anyway, so this may end up moot. Thanks to both of you.