If it’s a cash settled future, my intuition would be to let it expire. You get guaranteed savings on the trading costs of one leg. The gap is zero expectation, unless you have a prior about it’s bias (in which case you should make a strategy out of it).
My experience with letting VIX futures expire was not very positive

My system missed some rolls earlier this year (due to a combination of technical issues). One of the months early in the year, the gap between the last trade price (I believe it was 11.5) and the settlement price was over 1 point (I believe settlement was at 12.65). Since I was short, that lesson cost over $1K per contract

Just to ensure that I had properly learned the lesson, I "audited" the course again a few more months (although the spread was never as large again).
(Yes, I do trade the front VIX contract despite the fact that GAT advises against it. My excuse is that when I initially "calibrated" my system I haven't noticed at that time that GAT recommends the 2nd VIX contract and I've been too lazy to recalibrate. Actually, I've been working on an improved approach to trading Volatility -- and since this "improved" approach has been "only a few days away from being complete" for the last 6 months, I didn't want to take the time to recalibrate the existing system
A bigger issue is that it seems to me that the VIX settlement is being manipulated. There was a paper listed on Quantopian (March timeframe I believe) about the manipulation of the VIX settlement -- apparently its pretty doable due to the "cheap" deep out of money options that go into the VIX calculation and the fact that the "settlement" takes prices from only about 1 minute of time -- so the manipulation does not have to be long lasting. To your point about having an expectation about this and creating a strategy, I have thought about it. I haven't looked into it though mostly because I'm scared
I have not noticed a similar spread between last trade and settlement in other markets where my system missed rolls, however any delay in closing positions that result from expired futures contract can pose significant risk. On my end, while my Futures trades are automated, any "cleanup" of currency positions due to expired contracts has to be done by hand. Since I am lazy, it usually takes me a while (several days) to get around to this. Murphy's law usually kicks in the meantime -- I was in the middle of manually entering orders to close the FX positions that resulted from expired short GBP and EUR contracts just as the FED announced a rate increase (May or June timeframe). Surprisingly, the dollar actually plummeted due to the FED commentary released along with the rate increase. The result was another "cattle prod" moment.
Since the question of the "best" method has been posed, I figured that sharing some personal experiences with one of the approaches mentioned would potentially be helpful.