Please note that in this case, conversions with both search and display clicks on the path will be duplicated (they will appear in both Google Ads accounts).

Of course, Google Analytics will deduplicate the conversions and help you to see the entire conversion path in Multi-Channel Funnels reports.

You will be able to see the degree of overlap and decide how to take it into account, for example by adjusting the CPA/ROAS goal.

If you use conversions from Google Analytics instead of native Google Ads conversion tracking, you make it even more complicated.

Splitting the accounts won’t change the fact that you artificially decrease the volume of conversions visible for Google Ads.