Unfortunately it's not possible without paying Apple as far as I know. However, since Tiled is open source you could build it yourself to use it on your work Mac. There are instructions in the README.md.
Indeed, it requires a paid developer account (~100 $/year). Is that a blocker?
Well, it's hard for me to justify paying Apple to get its operating system to trust Tiled. No other operating system requires such a fee. Maybe it would be worth it if I was selling software on the App Store, but Tiled is free. If I would charge for it on the App Store I would be misleading users into thinking they need to pay for it, and Apple would screw me over further by taking a whopping 30% cut.
If enough of my supporters care about this I could consider paying the $100/year fee. Until then, since Tiled is free software, you should be able to compile it yourself by following the linked instructions.
It's a little easier than compiling from source to run unsigned on OSX (for most people; this might not apply to the work laptop case from the OP, but in case anyone else finds this...)
You can also run the precompiled software on osx by opening Finder > Applications (or whereever you installed it), right clicking the Tiled icon > left clicking `Open`, and then telling it "yes I really meant to open this software". You may have to try running it twice for the OS to give you the correct dialog, after which it's trusted.
Note you *can't* open the app with a doubleclick or the return key to get this dialog the first time (the normal "run this app" gestures; more details).
You don’t even have to compile it, at least not on Mojave. You just have to set a setting in security and right click on an application to open it
Can’t speak for everyone, but I would love to buy Tiled from Mac AppStore. Downloading from elsewhere is always bit troublesome and so it building the project from github. The hard part I guess would be the pricing.
It’s $100/year not per month. Still, I understand why some devs wouldn’t want to pay that fee