When you do Automate -> Batch, you refer to a certain script (or "action" in the Photoshop parlance). Just make that action explicitly save the image the way you want.
Open a sample image.
Create a new action and start recording.
If you want to export in a certain profile, do Edit -> Convert to Profile explicitly.
Do File -> Save As..., set all the options you want (set Format, tick ICC Profile, etc.) Better if you don't touch the file name: this way only the path will be recorded. It won't matter for a batch, but this way you could use the action as a one-off to save the current image with the original file name. If the format is the same as the source (e.g. jpg), pick a different folder so as not to overwrite the original. Then actually save it to record the action (and to select the format saving options like the jpeg quality).
Stop recording.
You should end up with an action consisting of one or two steps, the last being Save. (Of course, you can add more common actions before save if you want, like resizing, output sharpening, etc.)
Now select your action and invoke Automate -> Batch and set these options (you need to do it only once unless you change them again).
Check that your action is selected.
Your Source will presumably be Folder to process a batch of files, but use whatever is appropriate.
In your first screenshot with Color Settings, there are three checkboxes regarding the missing/mismatching profile. You don't have them set, but I always advocate to tick all of them: you do want to know what's happening when you work in a colour managed workflow. If you do have them ticked, you'll need to check Suppress Color Profile Warnings here in the Batch window.
Select Destination: Folder and choose one where you want your results to be saved.
Tick Override action "Save As". This will cause the files to be saved with the name and path as specified here in Batch, yet still with the selected options in the action.
Select Document Name + extension pattern (this is the default), or whatever you like.
Proceed with the batch (click OK).
A side note: jpeg is not a suitable format for ProPhoto RGB: 8 bit colour resolution is too small for such a wide gamut. You lose more than you gain. If you want to store an 'archive copy' in ProPhoto, you need a 16-bit-capable format. Alternatively, store the original (presumably RAW) + non-destructive editing in XMP, and the final jpg in sRGB or AdobeRGB.
