Need to measure high resistance values
15kOhm is well within the range of common multimeters. I did look up your particular model and noticed that there is no spec given for >10kOhm, so you may be correct that your meter is not designed to measure these impedances.  See page 23 here.

Here is an example of a low cost multimeter that could fit your needs. If you are interested in another Fluke, the 115 looks like a reasonable option.