How should I prorate property tax with a fiscal year different from the tax year on a new rental?

I just started renting out part of my property last year (2023), and while I'm letting Turbotax calculate splits due to the rental only covering part of the year (that computation looks correct), I'm wondering if I need to further pro-rate the property tax.

 

There are two reasons I'm not sure:

1. My city property tax bills run from July 2022-> July 2023. So, should I expense the actual payments I made in 2023, or pro-rate the total bill for 2022 -> 2023 to the days it runs through in 2023?

 

2. My home was purchased in 2022 and it took a while for the city to reassess the value for taxation. This meant in 2023 I received and paid extra tax bills to retrospectively cover 2022. Can I include these as a rental property expense for 2023?