Hello Guild Members, I am working on a world map that contains several layers such as coastlines, rivers, cities, overlays and name labels. Would you please advise me how to reproject it so that the layers are in sync?

Details about the map:
- Vector map created in Illustrator to allow for rescaling
- Equirectangular (2:1), assuming Earth size for utility

The Goal:
- To be able to change projections to the most common ones such as Winkel Tripel, Mercator, Orthographic, Robinson
- To have the output in similar format so that an artist can rescale it and add details

Approach to the problem:
- Cartography is an area I am an amateur in, my knowledge is at the novice level. I am familiar with PS and Illustrator but not with dedicated mapping software
- I tried G. Projector as it seems beginner friendly. However, it is not fit for the purpose as far as I understand. Also, like others in the forum, I found it to be limiting in file size
- I imagine there are programs that can reproject while keeping labels positioned to the correct object they describe. I guess that working in layers is typical for such programs?

What would you suggest?