You could certainly do it in an E. coli bacteria tomorrow. You'd need on the low end 128 or 256 bits of coding capacity, which you should be able to get out of a few thousand base pairs according to their scheme. Not super easy, but I've made strands that long using only oligos and with a little tinkering you could correct any errors. Now tie that DNA to an antibacterial-resistance gene, stick it in a bacteria, and you can selectively grow only PGP-bacteria nearly indefinitely (with some minor caveats, but yeah).

Houseplants are substantially harder for a variety of biochemical reasons including having multiple chromosomes, and more difficult selection protocols, but yeah that should be possible too and made somewhat easier by CRISPR.

Of course you could, but now that we know you plan to hide you PGP keys in your plants, you're hit by the old "security-by-obscurity" fallacy. :/

Just xor it with a one time pad, encoded in a fern.

Yes, you can. Depending on the organism type, it will shed that dna after some generations though - organisms with useless dna chunks have a slight evolutionary disadvantage.

I imagine impementing some kind of additional checks - e.g. special genes that kill organisms wih a bad checksum of the stored data.

Supposedly, assuming that the sequence doesn't harm the organism (or that the organism doesn't alter the sequence). Bacillus spores can remain dormant for decades, if not centuries, and date palms have been grown from millenia-old embryos.