if the slate requires just a top up of the stain stop, de grease with a light cleaner - poss lithofin power clean dilute 1-10 and test, leave to dry fully. this can then be topped up with the stain stop again.
if you want a more polished finish, i would test cobble wax after the st stop , check for colour and suitability . this will give a deeper colour and a rich shine to the surface- slate seal is to thick for this soft honed finish of stone and could cause problems with scratching and may not key correctly. other than that the client could use care sheen occasionally and bring the sheen up slowly as part of the maintenance.
The other floor , flag- this again clean back with either power clean or wexa - leave to fully dry ( this would be a few days for me) test using colour enhancer (1-2 coats) and then a coat of MN.
if you want a sheen finish on this stone -test lithofin multi-seal on top of these and this will give you a satin finish (also darken the stone slightly)- this will also be easier to maintain. ensure clients are aware that they will have to top up these finishes if they want them and maintain with the correct products .
For ref; if you wish to strip out (say to apply slate seal)- test wax off
no recommendation for steam cleaning with topical finishes;