So since this turns out to be specifically genetics:
To speed things up one would need a very huge population. Also one would need a very controlled environment to ensure all plants have the exact same condition to begin with, including water light fertilization and potting mix. To get even better sample size and address remaining variation in conditions several clones of each individual, to determine their average performance, would be even better. This leaves you with even greater population.
Since mutation occurs at random and has nothing to do with environment conditions (excluding mutagenes) speeding it up by use of mutagenes appears sensible.
Now: 99.9% of mutations will be negative, leaving plants crippled or making them less desirable in any way. Or in other words: Moving them further from the original without adding any benefit, let alone the desired change.
Of the remainong 0.1% beneficial mutations again 99.9% will not be what one is after (increased cold hardiness).
Add to that, that a plant may have several mutations, some good some bad, some even of the desired type. In such case some traits will have to be bred out, taking care the desirable mutation persists. Then the population will need stabilizing.
In the end cultivating any plant on a large scale and watching for any beneficial mutations will be a lot less frustrating than going for something very particular. Of course, if someone is determined to go the later route they might also go the extra mile and breed several variations with other benefits along the way.
Bottom line: Without a lot of spare time, dedication, tolerance to frustration and legal tender this task is out of reach for the hobby gardener. Making progress by chance, not intention, seems more likely to happen.
Of course there is the more scientific way of genome editing. As mentioned before: One would determine the cause of death at the iteration, research ways that makes other plants immune to it, trying to apply this knowledge by genome editing (still requiring a huge population and regular breeding out unintentional 'editing') and making sure the plant retains its original character or at least the usefulness one is after.
It's a long way.