After trying quite a few things, I figured out how to achieve the desired result via the API. Posting what I found here in case it helps others with the question of "how to create a new DSA campaign restricted to ONLY using the supplied Asset Set of URLs".
1. Created the campaign with a DynamicSearchAdsSetting. However within the setting, only supplied the DomainName property and the LanguageCode property. Completely omitted the UseSuppliedUrlsOnly property and the Feeds property.
2. Either created a new AssetSet of PageFeed URLs or grabbed the ResourceName of an existing AssetSet.
3. Used CampaignAssetSetService to make the association between the new Campaign and the AssetSet. (Note: at this point, if you check the interface you'll find that DynamicSearchAdsSetting.UseSuppliedUrlsOnly is set to FALSE which is the issue that needs overcome.)
4. Retrieve the DynamicSearchAdsSetting and change the UseSuppliedUrlsOnly from FALSE to TRUE.
5. Create a new CampaignOperation and set the Update to the newly altered DynamicSearchAdsSetting and then execute the operation. (Note: Now the setting in the Google Ads interface will be updated to only use the supplied URLs.
Note to API team. This would be much more streamlined if we could set the UseSuppliedUrlsOnly property in "STEP 1" of my flow (and still leave the Feeds property blank) without receiving an error.