Measuring self-capacitance of coils

Arguably, the easiest way is to just measure it:

      +--------+          
      |        | 
      |      [50R]
      |        |
      |       [C]
    [GEN]      |
      |        +------+
      |        |      |
      |       [L]  [SCOPE]
      |        |      |
      +--------+------+ 

With \$C\$ known, adjust the generator's output frequency for a peak across the scope, then determine the reactance of the capacitor with:

$$X_c=\frac1{2\pi fC}$$

Next, since the reactance of the capacitor and the inductor will be equal at resonance, (the frequency at the maximum amplitude of the peak on the scope) solve for the inductance with:

$$L=\frac{X_c}{2\pi f}$$

Both equations can be combined:

$$L=\frac{1}{(2\pi f)² C}$$

You may need to adjust the value of the \$R\$ in order to get a nice peak.

Regarding the bounty quest for reliability in a cheap way: to convince yourself that this is a rather difficult task to do reliably (with laboratory precision), have a look at what it entails to measure "it" for a human, e.g. in a paper that studied it for ESD-related purposes, Numerical Calculation of Human-Body Capacitance by Surface Charge Method by Osamu Fujiwara and Takanori Ikawa, doi:10.1002/ecja.10025. Quoting from the abstract:

However, the body capacitance is strongly dependent on the relationship between the ground plane and the body posture. It is therefore not clear what factors govern the body capacitance. In this paper, the static capacitance of a body standing on a ground plane is calculated by means of the surface charge method. [...] It is found that the capacitance increases as the backs of the soles of the shoes approach the ground plane, that the body capacitance at the same height (10 mm) as the soles of the shoes is 120 to 130 pF, and that it is about 60 pF if the location of the soles is sufficiently high. The computational findings are confirmed by measurement of the body capacitance.

And if you're curious about their measurement method, here are the details for that from the paper:

Figure 7(a) shows the method of measurement of the human-body capacitance and Fig. 7(b) shows its equivalent circuit. The person tested (height 168 cm, weight 68 kg) with a body shape close to the human-body model is standing with bare feet on a Styrofoam plate or a perforated acrylic plate (depth 30 cm, width 11 cm) on a metal plate in a Faraday shield. The perforated acrylic plate has 201 holes made with a drill with a diameter of 4.5 mm at random locations over the plate and with an area ratio of about 9%. In this way, the relative permittivity is effectively decreased. Under this condition, a power supply is used for charging to V_B0 (= 10 V) via an analog switch (Toshiba TC4066BP). When the power supply is turned off by the analog switch, the body potential v_B(t) is amplified by a low-input-impedance amplifier (with an input resistance R_i = 10.2 MOhm, input capacitance C_i = 13.6 pF) and is directed to a computer via an A/D converter. The sampling frequency of the A/D converter is 200 kHz and the quantization level is 12 bits. In the potential measurement, the metal plate is used as the ground to which the grounding connections of the measurement devices are connected. From the equivalent circuit in Fig. 7(b), the body potential v_B(t) is given by

$$ \frac{v_B(t)}{V_{B0}} \simeq exp \Big[ - \frac{t}{(C_i+C_B)R_i}\Big]$$

Hence, from the potential decay characteristic, the body capacitance C_B can be derived.

This is basically the same time constant method suggested by George Herold (which I upvoted a while back), but at boffin standards. Nobody measures body capacitance with regularity (even for humans), so I don't know why you expect there to be a cheap way to do it reliably... Never mind that it would probably vary quite a bit as the cat changes body position.

Also, if you hope to just do simulate it on a computer... their numerical model likely won't wont be much good for a cat because:

In addition, clothes and hair are not included in the numerical model.

For a somewhat older (but right now freely available) paper, which discusses the problems with getting accurate body capacitance measurements, see N. Jonassen's Human body capacitance: static or dynamic concept?. Reading that, one point that was salient was that the soles of the shoes are actually a major contributor to the human body model capacitance (while hair and clothing can be basically ignored). Alas, that's probably the opposite of what you can expect for the dominant element to be in a cat (in its natural state) as far as capacitance is concerned. Unfortunately bounty points on SE are rather unlikely to be a sufficient "grant" for boffins to tackle this rather different cat body model in their labs...