Best bag for on a stroller for 2 kids for all day outings

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Need new diaper bag. We are going to Rome for 6 days. We will have a Phil and Ted sport stroller and need a diaper bag for days out. Packing for a 2 year old (still in diapers) and a 2 month old. Here are the options. Bonus if I can use it as the plane bag too!

Things the bag should hold:
2 sets of diapers/wipes
Sun screen
Snacks for 2 year old
Clothes kids need to take off/put on for layers
Water
Bottle/sippy cup for 2 year old and 2 month old
Wallet
Cell phone
Some toys/hard board books

Thanks in advance!