It's not really 'dating' as much as it's pre-established romance from Fate/Extra, but yes, you can - though as has been said, there's barely, if any, difference in what the characters say to a male avatar vs a female.
The female avatar still gets called husband by Tamamo, for example - though IIRC that's explained in Extra as just a foible of Tamamo always seeing herself as 'the wife'.
Also, glad I popped in, didn't know there was going to be a sequel.
Though I'm curious on how they plan on modifying Extella's storyline - if it's going to be similar to what they did for CCC, or an actual sequel.
Either way, I kinda hope they drop Vita in order to bump up the production values and some other less-than-stellar elements of the musou action (though they probably won't
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