How do i get Kana Birthday card??

[Ayu]

i can't get Kana's Birthday card, i found RIn's card and Miuka's card on the 1st and 8th day of July but i can't never find Kana's card before her brithday

[VertVentus]

I got my birthday invitation during my second Afters with Kana, as early as 6/24 since I'm savescum gambling on my first playthrough.

https://twitter.com/NAl0lAN/status/1781547398864764940
・You will be invited to a birthday event after the event if your favorability rating is 4 or 5.
(Sometimes only Kana will give it to you after the first time. Rin and Miuka will not give it to you until the second or third time.)

[NuclearRaven]

Originally posted by VertVentus:

I got my birthday invitation during my second Afters with Kana, as early as 6/24 since I'm savescum gambling on my first playthrough.

https://twitter.com/NAl0lAN/status/1781547398864764940
・You will be invited to a birthday event after the event if your favorability rating is 4 or 5.
(Sometimes only Kana will give it to you after the first time. Rin and Miuka will not give it to you until the second or third time.)

This isn't true.
Miuka gave it to me at favorability level 1. I forget when Kana gave me hers but I've only got her at level 3 in this run. Meanwhile Rin at level 7 still hasn't handed over hers.
It definitely happens at the end of after party events, but requirements for them to hand it over appears a bit random.

[VertVentus]

I did notice some discrepancies with their list of days off, and this other wiki corroborates what you said, so you're probably right.
https://www.gameludens.com/bunnygarden/154/
"If you hang out after the party, you may receive a birthday invitation ticket. Miuka got it the first time, but Kana got it when her favorability rating was 3, so at least she didn't get it the first time. If you go on your birthday, you can also receive a present, so don't forget to go there."
https://www.gameludens.com/bunnygarden/393/
"I've played there a few times, but I often get birthday tickets after the event in July, and it looks like I won't be able to get them in August. My understanding is that as a provisional condition, I will go to the after-sales service in July."

The birthdays in order are Kana on 8/13, Miuka on 8/27, and Rin on 9/3.

I got my first Kana afters on 6/4 at 4.5 likability, then her second on 6/24 by 9.6 likability. Maybe the chance of getting an afters increases with how much you spend?

[NuclearRaven]

Well, got Rin to affection 10 got multiple afters including 2 luxury afters and...she didn't invite me to her birthday. Dunno about you, but I feel like this shouldn't be possible. Probably needs patching, but I guess I'll try for it again in the gold mode

[VertVentus]

From the Twitter post I linked first,
・If I raise my favorability too much, I can't get invited to a birthday event?

My first afters for Rin was 6/4 (6 likability), second on 6/18 (7 likability), third on 7/1 (8 likability) and that was when I got her birthday ticket, corroborating what OP said.
So maybe savescum gambling or gold mode can actually screw you over, you'll want to spend Yomands deliberately on your solo girl playthroughs. I only spent 7 up to this point and stopped gambling.

For Kana, I got her 3rd afters on 7/9, 4th on 7/23 (swimsuit), then 5th on 7/29 for the vacation trip. There were more after that on 8/6, 8/20 & 9/10 (nothing) and 9/17 for the confession.
I found out about that 8/6 afters on a "second" Kana playthrough, redoing my save to extract more conversations since I used her vacation on a bad weekend (8/5~8/6 when she has 2 days off consecutively on 9/2~9/3), so while the sequential order of conversations got arranged and I got new conversations for 9/16~9/17, I'm thinking that afters are set dates.