Getting and providing evidence of a third shot / booster for a foreign resident child

Came to Japan from Australia when my kid was 11 and had/has two jabs. She’s now 12, we will be going to England in a few months for a holiday and I’m thinking ahead to the re-entry to Japan.

Looking at the new ‘you don’t need a negative PCR 72 hours before flight as long as you have 3 jabs’ rule, I presume that as she is 12 (with 2 jabs where 3 are permitted) that ruling would not apply to her. I want to comply with that rule by getting her a third jab / booster now (it’s been over 6 months since the last)

I have two questions:

Firstly how would I get her a jab? Googling reveals that they send you a voucher. (Although not really sure who they is). In Aus you can walk in to any pharmacy and get one – is there some thing similar here?

Secondly, her record is with the Aus government, who provided an international certificate that was used for entry. Would the Japanese add to that existing certificate or create a new one or maybe have a Japanese cert that says only that she’s had one.. or go back to Australia and ask them to re-issue?

My outcome is simple – to come back in without needing a 72 hour before flight negative PCR (fell foul of that one and it caused quite a bit of difficulty).

Thanks