You don't need a student VISA for hour building. I have done this many times (I have a FAA PPL based on EASA PPL). The BFR isnt training because it doesnt count as credit towards a certificate or qualification.
To avoid hassle at immigration I've refrained from saying I was there to hour build (which implies training) but instead said that I was there on vacation and that I intended to rent an aeroplane. Which I did. I went all over the north east on a grand tour with my family.
One final bit of advice. If you are in doubt get a lawyer to figure it out. If you get into trouble and then refer the judge to a bunch of unqualified people (in legal terms) talking crap online will not help you. If you got legal advice and it was bad you have comeback.