Electronic – Measuring resistance with ohmmeter over resistor and capacitor in parallel

You can measure Rdson under the conditions that you specify, BUT your measurement conditions will be so different from what you would normally experience that you will need to be very careful to ensure that it tells you what you want to know.
Most meters use a 9V battery (6V-9V) but some use 2 x AA = 2 to 3V and some use 1 x AA (rare).
Meter current may be 1 mA or some other current. The lower the test current the less the test will be like most real world conditions. If your Itest is 1 mA and Icircuit is going to be 10 mA then results may be quite different.
Ambient temperature will affect results.

You may have a good reason for doing this rather than by using the data sheet but it is not at all apparent what useful purpose would be served. You will get a one off result in a very special and uncertain circumstance that may be quite different than what you will get under similar circumstances subsequently.

MOSFET Rdson is dependent on Vgs. For Vgs more than a few gnat's whiskers above Vth - the gate threshold voltage , Rdson will be about as low as you can get for that device. But if meter current is say 1 mA and you increases Ids to say 10 mA you may* find that Rdson increases very substantially.

Below are Ids versus Vgs curves for a 2N7002. It can be seen in the left hand graph that for Vgs = 10V the Rdson is about R=V/I = 0.85V/0.4A say ~= 2.1 Ohms. For high values of current curves for lower Vgs curve off to the right - ie Rdson increases markedly above a certain Ids in each case. BUT for a low enough Ids the curves all tend to asymptote to the same as the Vgs=10V line. It may not appear that the Vgs=3V lines does this but that is mostly due to the scale of the graph axis. The right hand curves show the relatively linear behaviour of Rdson with Vgs at a fixed temperature down to Vgs ~= 3V at 25 degrees Celsius.

An alternative to relying on relatively undesigned meter based voltages and currents is to place a resistor Rd in series with the drain and apply Vgs as desired then measure Vds and (Vsupply-Vds) = Vresistor.
Then Ids = Vresistor/ Rd.
Vds = measured. Vgs = as set by you. Then Rdson = Vds / Ids.

A technically superior alternative is to set up a variable power supply to current limit at design current - say Ids = 10 mA, and apply Vgs as desired.
Measure Vds and all done.
Rdson = Vds / Ids

If you do use the meter method then adding a series resistor Rd will allow you to both measure meter current and to check the result as above.
You'd need a separate voltmeter.

Is it be correct to say that the CSR = ESR + external resistor = 1.3 + 0.5 = 1.8 ohms?

According to a footnote in the datasheet, "CSR(compensation series resistance) refers to the total series resistance, including the equivalent series resistance (ESR) of the capacitor, any series resistance added externally, and PWB trace resistance to C_O."

In most cases, the trace resistance will be small enough that you can ignore it, and your formula would be correct enough.

Now I would much rather use a ceramic cap, which from what I've read, has an ESR in the milliohms.

Remember that a ceramic cap in the 10 - 15 uF range is going to be fairly large, which can cause reliability issues (due to mismatched CTE between the cap and board).

It looks like a CSR of 0.1 to 2 ohms would be sufficient.

Notice, comparing figs 28 and 29, that the required CSR changes depending on the capacitor value.

And your capacitor value will change depending on voltage and temperature. But 1-3 Ohms total resistance seems to be okay for any capacitance value.

If I use a ceramic cap instead of the tantalum, and if I add an external resistance of say 1 ohm would this yield proper operation?

Based on the above, this should work. Be sure that your capacitor will have 10 uF or so minimum value over all operating conditions.