Here's a pretty full answer to this one as I got some decent questions in DMs (which are mostly open). It's not the only answer, of course, not the best answer, but it works. Start from a low dimensional model parametrisation, SABR or SVI or something you cooked up https://twitter.com/Mephisto731/status/1338992999544594432

SABR and SVI can both be restricted to be arbitrage free. I'm going to use SVI because I know it best, but both work. I always write SVI in the "jump wings" format which has ('vol', 'skew', 'put','call','minvol') as its 5 parameters

Convert from 'minvol' to ATF curvature as the 5th parameter. Just another language change, nothing complex in this. Now I gave only 2 points there's no justification for any curvature - so set that to zero. Down to 4 parameters.

For both 'put' and 'call' parameters, these are bounded > n.skew for some n. Rewrite them as multiples or spread over the skew bound. Fix that at some reasonable level

'Reasonable' depends on other assets, historical data for this asset, what you has for breakfast. Whatever. Basically now you have a 2-parameter smile model where your params are only 'vol,skew'. Easy to fit those to the 2 points i gave you.

With reasonable choices for the wing bounds you can prove your smile is arb free within some skew range, and you're done.

The worthwhile game for professional vol quants here is not to build a 16 parameter model that you throw at everything. It's to build a family of models that you can restrict/extend from 2 to 16 parameters as needed.

This was an example of going down that ladder. Up the ladder is naturally more intricate.

For this low information case there's likely to be a maximum entropy approach that you could use directly with the probability distribution. I've never tried