I don't think you can have Kitty with Tabitha, sorry. In my class one year I had two Kittys, a Felix and a Rex - all nice names but put together I felt like I was working in a pet shop!
Posy is cute. Not sure about on an adult but I don't know that it's really a short for anything. I know one and think she is just Posy. It will probably come to seem as normal as Poppy and Lily have so will be fine.
I don't really like Lola. Not sure why I think it's the combination of 'oh' and 'll' sounds.
If you like names in that vein have you considered (as either short versions of other names or standalones):
Annie, Abbie, Allie, Bonnie, Bessie, Betsy, Bronte, Brodie, Bridie, Briley, Cassie, Callie, Carrie, Casey, Chloe, Demi, Darcey, Dixie, Dulcie, Ellie, Evie, Edie, Esme, Elsie, Florrie, Georgie, Gracie, Hallie, Hayley, Honey, Holly, Hebe, Indie, Ivy, Jessie, Josie, Jody, Kaylie, Kerry, Kelsey, Kenzie, Lily, Libby, Lottie, Lexie, Lacey, Livvy, Lindy, Lizzie, Molly, Millie, Macie, Miley, Marley, Marnie, Maddie, Nancy, Poppy, Polly, Peggy, Penny, Rosie, Ruby, Riley, Sadie, Shelby, Sydney, Susie, Sally, Sylvie, Tilly, Tansy, Trixie, Vicky, Wendy, Xanthe, Zoe.