The problem is that it's not so much a "phase" as him demonstrating a fundamental need; cats are highly social creatures by nature. It's like asking if a cat will "grow out" of being upset when sprayed with water daily; the point is that you should not put a cat in that situation. They may "grow out" of demonstrating it outwardly, but it's not healthy at all for their mental state.
Doing a bit of research,
you can ask for a second ESA letter for the other cat; your original cat needing company while you're at work seems like a reasonable motive to ask for it. I think a landlord would be rather petty to object to two cats given a good reason, if they're willing (or forced to) to allow a cat in the first place, but you know... landlords. Perhaps negotiate with them, e.g. offer a deposit for any potential damages the cats could cause?
More elaborate alternatives would include either someone coming to your house to take care of your cat while you're at work, or you dropping him somewhere that can care for him. This is obviously very complicated both for you and the cat even without taking COVID into account.
Temporary bandaid solutions include anything that can distract him in the meantime: toys of all kinds, a cat tree set up by a window to watch the street, perhaps leaving the TV on if he pays any attention to it, etc. Also try scents that make cats more at ease, like valerian herb (spread a bit of the herb around the house) or Feliway. I would do
all of the above and see if anything helps, but again these are at best stopgap solutions to a fundamental underlying issue.