That would be better, drxx, if 'test-drive' were a noun. Chambers has it as a verb – and a transitive one to boot. So it would not work grammatically and would still leave 'with' as a redundant word.
It seems that the surface reading is meant to convey the idea of a driving test so a better wording would have been 'In Jeep I do test for examiner' where 'for' can be be interpreted to mean 'in order to obtain (a word meaning) examiner'.