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Short, easy-to-read gene tags.

Since all modifications are reliably policed, you can require that every modification includes a short, non-coding DNA sequence containing the recipient's legal identity.  For good measure they could add the date, the serial number of the clinic, and maybe a regulator-provided hash for security.  It could be done in less than a hundred DNA bases unless you want a really long hash.

How quickly and cheaply the tag can be read depends on the nature of your "extremely cheap and fast" sequencing technology.  Whatever it is, you don't need to read the entire genome. The tag could have a standard sequence at each end for PCR amplification, and if that isn't enough you could use multiple copies or put the one copy at a standard point in the genome.  Some current instruments do parallel sequencing to keep the total time down, so it's conceivable that your advanced sequencer might take as long to read a short tag as to read the entire genome, but not likely.

I suggest a testing station in every town where people go to have their identity verified every time they renew passports, driving licences etc.  These documents (as now) include a recent photograph, with the requirement that it be renewed if a significant change is made to the person's appearance.  If someone gets a "new" face coded, they might have to update their documents a couple of times while it grows to shape, adding to the cost of the treatment.  If the testing is fast enough, airports can be equipped to compare people's current DNA to their passports before they fly.

The extreme version would be a cheap portable device that grabs a few skin cells and reads out the identity a few seconds later, negating the need for other documents,  but that would raise several privacy issues.  It would also be open to abuse - gloves impregnated with someone else's DNA, dodgy practitioners modifying only the skin of the hands...

Non-modified individuals can be IDed in traditional ways, but as you're reading DNA anyway, any known highly-variable region could be recorded and used as ID.  Since illegal changes are always reverted, it might be that the authorities already keep a record of each individual's original genome - how else would they know?