I live in a small town outside of my target market, but Google maps will not let me show my business in the larger market because my business is based on my home address. Is there a way to work around this?

Rating: +1

You can target your ads by geographic region
without using your business address.

To do this select the campaigns you wish to change the targeting for
and click Edit Settings, then at the bottom of the right hand column
under Target Audience click Edit Location. That will open up a
new page and display your Current Location Targeting, click the
Change option on the right side of the page. That opens up the page
that allows you to select targeting by Countries and territories,
Regions and Cities (what you are looking for) and Customized
targeting.

If you do want to continue running Local Business ads on Google Maps
you might consider getting a PO Box that is in your target market.