Rumor: Supergirl to guest star in Season 1

yes, I can confirm this isn't just a rumor. However, how Supergirl will crossover and when is being worked on. If the likely outcome of Crisis is that Earth 38 and Earth 1 merge, then post Crisis crossovers will be easier, at least story-wise.

Treat this as a rumour and nothing more until it is officially confirmed. Batwoman may be extended past its initial 13-episode order provided the ratings are steady and there is sufficient public demand for it. Many networks, especially for new shows, do a 13-episode initial order and if it proves to be successful, extend them to a full 20 to 22-episode season.

As for Supergirl appearing on Batwoman or vice-versa, it is a possibility. Just like the Flash did in S1 of Supergirl, a post-Crisis meeting is possible but it would likely occur after the CoIE story is complete. So, all said, it would not in all likelihood happen before spring 2020.

For me, it's more than a rumor that Supergirl and Batwoman will be joining forces in more than "just" the crossover. Among others, Caroline Dries (executive producer) was already pointing in that direction.
I guess it's not that easy bringing both together in one show with the tight schedules and all, but I am sure there will be more with these two together.
Perhaps they can make a "swap". Batwoman appears in one Supergirl episode...and Supergirl appears in one Batwoman episode
I heard about the "22 episodes" rumor. Well, I'm superstitious and I won't believe that, until it's confirmed officially, but it would be great if makers and producers have so much confidence in the show, that they greenlight 22 episodes before even the pilot aired.
Until then it's "step by step" for me. 13-->22 episodes, season 1---> season 2, ---> season 3 etc.
I really hope that this show will last as long as the others did so far.