Hello @Katare, Ashish
Sorry for the late reply but below are my thoughts on your questions:
Would Front door with Basic APIM would work in above use case or what am in missing for not having the premium APIM.
APIM as a service comprises multiple components that work together coherently in-sync in order to provide its features -- the gateway component is an essential one that's in charge of running your policies and transforming the requests/responses. With the Premium tier, APIM is providing an ability to deploy Gateway components to various regions of your choice while still letting you handle Management tasks from a single, central plane (and the same goes for the Dev Portal component and its users, etc).
So there's a one-many relationship from the management plane to your gateway instances, meaning that when/if you make configuration changes, they'll sync automatically to all of your regional Gateway instances (same thing applies for any Patches and Feature Updates that APIM provides over time as well). If you go with the Front Door & APIM Basic approach, you'd end up with multiple Management plane & Dev Portal components as per APIM service instance basis which is not ideal.
What if I decide to have traffic manager along with Basic APIM instances - would that also work or what is preferred?
Same scalability issue for the management & dev portal components.
I hope this clears it up a little bit but feel free to let me know if you have any further questions on this matter.