Haha, sorry no. For one that is against the EULA I for one am a Trek fan from "Way back when" and I love flying my Konnie and Vengeance (though I hated Into Darkness) Already bought my Tickets for Beyond based on the reviews I've read.
CBS won't let them. The Kelvin Timeline Connie isn't the TOS Connie, which is why it was allowed. (Also lore reasons say it is more advanced then the TOS connie)
Make sure you've completed or skipped every mission up to and including the Solenae Dyson sphere arc And also check under the available missions/contacts tab.
This has nothing with the mission really, but I don't think the Yorktowns 'phasers' fire fast enough, if you look at movie 11 they're just a flurry of bolts, kind of like how the phasers were in The Wrath of Khan, just not as narrow of a firing arc.