Not exactly what you're looking for, but similar things...
In the Watch app on your iPhone, in Clock > City Abbreviations, you can change the name of your local timezone to "", and then have a complication that shows an Apple and the local time, and you can change your monogram to "" in the adjacent settings and get that to show if you use the "Color" face.
I actually set the local timezone name to nothing, so I can have an uncluttered digital local time complication on the analog faces, but i almost always end up using the Modular face.
if you can't get the actual "" symbol, you could copy/paste from this message, or google for "apple logo symbol", or, I get arbitrary Unicode characters from an iOS app called "Unicode Pro" (type "apple logo" into its search field).
(Technically, the Apple logo isn't a proper Unicode character, it's a "private use" Unicode character that Apple devices will show as an Apple symbol. If you read this post on a non-Apple device, you may see three instances of some completely different symbol, in quotes, up above.)