I've only once played with hermaphrodites in an experiment to make female seeds, because its female seeds I was after I culled all "normally/naturally" male plants.
The reason behind this was simply that I did not believe that hermaphrodites can produce only female seeds, just that the pollen made by hermaphroditic plants is missing some genetic information required to create "normal/natural" male seeds. Therefore in my understanding your hermaphroditic pant can be pollinated by a "true/natural" male and produce some male seeds, however having its own pollen right there (all the time) I find it unlikely that you will be able to exclude enough of her own pollen to yield reliable results.
Here's the biggest Issue I see... If you do successfully outside pollinate a hermaphrodite (which should be possible, just less likely) the only way you would know you had seceded is if you found a male plant amongst the seeds (and what good does that male do you?). The very next seed over in the same bud could very well be born of the hermaphrodite pollen..
Another thought/perspective is simply that seeds pulled from your hermaphroditic mother are the only ones you should have to worry about being "genetically weak" and the only reason is that every recessive trait is twice as represented in the seeds pollinated by herself (which could work against you). Its pollen from hermaphrodites that is used to make feminized plants (atleast was when I did it).
We have all seen feminized seeds grow some nice plants at this point (though I don't believe seedbanks sell the seeds directly from the hermaphrodites bud).
Here's the interesting bit if you hatched out those seeds there may be a chance that you will encounter a higher proportion of mutants and recessive traits, however some might express her most desirable traits too. If you have that situation and you want to stabilize that trait in future offspring (stabilizing the strain/phenotype) what I think you would want to do is obtain some "regular" seeds grow them out and select the most robust male with the stature you are going after and breed them. (you may want to preserve cuttings from the "favorable mutation mother" as you may want to back cross her into subsequent generations, if her most desirable trait thins out in subsequent generations (It could be a recessive trait after all).
Keep in mind this is all speculation... I have not begun a serious breeding program yet (only played with feminized seeds years ago because i didn't believe the roomers floating around at the time). I am sure there is alot I have to learn to become as successful a breeder as I would like to be some day..
There are some threads around here exclusively dedicated to breeding, I bet there are a few guys/ladies around those arias that could offer alot more incite (and first hand experience) than I can..
There's a guy here that spoke of
useing Colchicine to produce polyploid plants. which could be really interesting if those results were expressed in subsequent generations from breeding a female with a hermaphrodite that have both been manipulated in this fashion, it might help overcome concerns about "genetic weakness" Most interestingly to me is that it seems you might be able to fortify a "freak" so that its trait were more consistently expressed in subsequent generations "make it more stable" (mind you this is all speculation and I have not looked into it at any depth, I'm not simply not "there" yet).
Female and Feminized Seed seems to touch on it a little.
Good luck, and don't forget to have fun