I don't think the client API has anything to do with it. The plans above have the same shape, except that that the plan for SqlClient has some extra operators due to the data-type mismatch. Those operators are not likely to cost you 50 seconds, though. There can, however, be other differences not revealed in the graphics, but we would need to see the XML for that.
I would guess that it is a parameter-sniffing issue, so that the ODBC client gets a cached plan optimised for a different set of parameter values than the one you have now. And by some chance, that plan is faster than one optimised for the current values.
ODBCClient and SQLClient gets different cache entry since the text is different.
If you want general help with optimising the query, we need to see the view definition, the definition of the underlying tables with their indexes and the execution plans in XML form. Since you mask names in the graph, I can understand that you may not want to do this in a public forum. But we can help with optimisation without facts to work from.