We also had to cancel our April 2020 trip (with stopover in Chengdu) for the same reasons. Our re-do is this August but we are flying to/from Tokyo directly from London. Alas, as our kid is now 13 she’s no longer elegible for child travel fares.

At age 5 our kid could outlast us all mountain hiking, she had rather less stamina in the cities. We found her Maxi Micro Scooter was helpful for some of our urban explorations, if she got tired we could extend the handle and pull her along. Late May/early June is already uncomfortably warm and will slow down your pace and what you can do compared to how you would manage on a March trip. Our Tokyo visit when my daughter was 5 was early on in May just as that humid warmth was starting to set in, the scooter did help her (and thus us) with covering more ground and for longer than she might have been capable of otherwise.

When she was 1yo my daughter was almost exclusively carried, passed between whichever adults were with her. As mentioned when discussing Mise Mountain and Mount Takao, if you are relying on a stroller there will be problems and hurdles that won’t exist if you are baby wearers. And there are other difficult places too, for instance, the main walkways to Meiji Jingu are channels of gravel, and some of the Japanese Gardens will have paths that are a major challenge for buggies/prams.
You may need to find out if the stroller you’re taking can be stored in taxi boots. Also, using subways and trains with a buggy will force you onto the elevator routes which will significantly increase the distances you need to walk. If you are buggy users you will probably have a platform the older child can stand and ride on. If you haven’t I’d recommend one (which might be easier than managing a buggy AND minding a scooter)

For late May, early June, even more than with the abandoned March trip, I suggest you take it slow.

A full day at Showa Kinen Koen with swimsuits and towels for the paddling pool areas and the chance to hire bicycles with child carrying seats to help you cover all the gardens would be delightful.
The Rainbow Pools which are for swimming will be closed (generally until mid July) but the Water Playland paddling pools and “beach” area will be open. Not much information about them, here’s a photo taken early in May with LapGirl who spent hours in this area being completely independent of us and having a great time. UV is stronger than we are used to (and I’m half Spanish) hence the little suncloak. You asked what to pack, I say whatever you need to keep their exposed skin safe, the kids won’t want to be wearing much clothing.