Hello tonycstech.
The UT181 will put a max compliance voltage of 1.2 in the kOhm range and upwards, below that is 3.3V. The graphing is not live though, you have to record and then plot, unless you use the included IR interface.
The Metrix MTX 3282 (and I guess the rest of the series) has an around 1.2V max compliance voltage, which drops pretty quickly. It is discontinued, and a bit quirky in a few regards, but you might find one second hand. It does live graphing.
2N3055 recommended, in the third reply, a currently produced model. Also has live graphing but it is quite expensive.
Also, the Ohms function in DMMs is designed to work with ohmic circuits, and intends to give results relative to Ohm's law. If you are using it for a different purpose and it doesn't work, it doesn't mean there is a design flaw, or that the meter is crap, you are just using it for a different purpose. The fact that some of them work is just a nice bonus with some DMMs. That's why you are getting flak.
I think what you are trying to measure is a circuit signature (not sure about that name). There are expensive curve tracers for automated test setups, which compare the traces of products with a known good sample in the production line.
In this case, that is kind of what you are doing, identifying problems with controlled test conditions by comparing with a known good sample. You might be able to use the ohms function, but the tool for this is not necessarily an ohm meter.
You might be better off modifying one of the "shorty" circuits, or using an Analog Discovery.
EDIT: The diode test voltage of both the Metrix and the UT181A is ~3.1V.