You should have a look at the article "Trace Paley-Wiener theorem for reductive p-adic groups" by Bernstein, Deligne and Kazhdan, which certainly gives the test functions that you are after. Of course there is work involved in getting an expression that is a simple as possible for the geometric side of the trace formula (if you pick a minimal parabolic, I would not be surprised if there was not much simplification possible). Perhaps there is something particularly nice in the case where the chosen parabolic at the chosen place does not arise from one over the global field. I am not aware of any article doing this.

You should have a look at the article

Bernstein, J.; Deligne, P.; Kazhdan, D., Trace Paley-Wiener theorem for reductive p-adic groups, J. Anal. Math. 47, 180-192 (1986). ZBL0634.22011.

which certainly gives the test functions that you are after. Of course there is work involved in getting an expression that is a simple as possible for the geometric side of the trace formula (if you pick a minimal parabolic, I would not be surprised if there was not much simplification possible). Perhaps there is something particularly nice in the case where the chosen parabolic at the chosen place does not arise from one over the global field. I am not aware of any article doing this.