Less-than-unity inverting op amp oscillating. Why

I think you have a few issues that you need to address.

The primary issue is that the Q of your tank is very low. The Q of a parallel RLC circuit is:

$$Q=R\sqrt{\frac{C}{L}}$$

Considering that your resistive feedback is effectively in parallel with the tank, we can calculate Q:

$$ 1500\Omega \sqrt{\frac{1nF}{15\mu H}}=Q= 12 $$

The Q of 12 is VERY low, so this is probably your main issue. Increasing the magnitude of R1 and R2 should be sufficient for your circuit to oscillate properly. See my example simulation here. If your Q is too low, you will need more gain to compensate for the losses.

A second potential problem is that the op-amp you selected, the 741, has a bandwidth that is a bit low for the frequency you selected. The datasheet indicates a bandwidth of 1.5 MHz, and your oscillator frequency is 1.3 MHz. This may result in your op-amp not providing enough gain for the oscillator to function properly. There are MANY op-amps that would provide an improvement over a 741.

Another possible issue is that oscillators are a bit tricky to get working in simulators. While it sounds like this is not an issue for you, it is a potential pitfall. Often, the random noise that usually starts oscillators in reality does not occur in a simulator. Often a noise source or impulse is required to kick-start the oscillator.

Without seeing the input signals, I'd say the circuit is working just fine. That is, it's working as it should given the topology. Unfortunately, it won't work as an instrumentation amplifier, or even as a differential amplifier. And I'm pretty sure a differential amplifier with gain is what you intended.

First, of course, is the fact that you're feeding an AC-coupled signal to a single-supply op amp, and this is guaranteed to clip all inputs which are less than the average. Are you sure you want to do that? Granted, it's a poor man's rectifier, but it's bad for the op amps.

Each amplifier has a nominal gain of 63. Let's assume the each has an input of 10 mV. Then the output of the lower amp will be 630 mV. If the upper amp were operating from a (very large) split supply, it would put out 630 mV - (63 * 620 mV), or -38.4 volts. This, obviously, is not going to happen. The problem you have is that your topology only works as a differential amplifier when the gain is 1.

There is a variant on this which supports gains >1

^{simulate this circuit – Schematic created using CircuitLab}

Common mode range is limited.

Another problem you have is that your input AC coupling networks are not obviously matched. Identical AC signals can produce non-zero outputs due to differential phase shift. Combined with the clipping produced by your AC coupling circuit, I'd expect some very odd results.