IMHO what you experience with your neighbour's strong signal is not intermodulation, but blocking / reciprocal mixing.
To generate IM, 2 signals are needed who's fundamental or harmonics combine to a mixing product that falls into your reception bandwidth, i.e.
n*f
1 +- m*f
2 = f
r, with n, m = 1, 2, ....
For third order IM products, usually n=2, m=1 and sign = -. The typical IM situation is 40m at night, where the strong broadcast signals directly above the ham band result in a multitude of intermodulation products that make the band feel "full".
You have only one strong signal, but a local oscillator with a lousy phase noise performance. What happens, is that your mixer "reverses" it's function, the strong signal works as LO and transfers the part of your LO's noise sidebands that is f
r - f
i away from your LO's carrier into your reception channel. What you observe, is not an identifiable signal, but just that the noise level of your RX goes significantly up when the "nice guy" transmits.
The SI5351 is a clock generator. It never has been conceived as a local oscillator in high performance receivers, und thus it's phase noise is not specified. The only value concerning short term stability that appears in the data sheet is "jitter < 70 ps"
While it is rather straightforward (albeit tedious) to calculate jitter from a given phase noise plot, the inverse is not possible, or only under a lot of assumptions (corner frequencies of the phase noise plot's segments). One on line calculator
https://www.changpuak.ch/electronics/phase_noise_jitter_conversion_reverse.php I found converts the 70 ps into -107 dBc at 100 kHz. Considering what I just mentioned, I read this as "somewhere between -120 and -95 dBc.
An AD9951 DDS chip, run from a clean clock source, should at 23MHz have a phase noise of around -150 dBc, -145 at least.
To find out, if what I briefly described is the source of your problems, disconnect your SI5351 oscillator from the mixer's LO input, and replace it with the output of a known low phase noise signal generator. An old HP8640 will perfectly do. If you are then able to receive weak signals some 200 kHz beside your neigbour, you should seriously rethink using the SI5351.
"normal" short wave receivers have Mixer IIP'3's somewhere between 10 (rather bad) and 30 dBm ( rather good). So, when you get 20 dBm out of your MC1496, you are somewhere in the middle, and that's not too bad. Actually, I am surprized this chip is so good, I still have a couple of them laying around...
With H-mode Mixers IIP3's of over 40dBm are possible. To achieve this in a receiver, not only the mixer itself on the test bench, EVERYTHING must be carefully selected. Besides the mixer, these are: The input band selection filters, the roofing filter, the diplexer terminating the mixer, up to the IF strip to avoid the in-channel IM behaviour being much worse than the out-channel values. You probably know martein, PA3AKE's work on this topic, if not, here is the reference:
https://martein.home.xs4all.nl/pa3ake/hmode/