The first thing you have to decide is if the data is miss completely at random, missing at random, or missing not at random. If its the latter, that is something about overnight stays itself is causing missing data, you are out of luck. There are no good solutions for that. For instance if as overnight stays take longer there is more missing data you have a serious problem (I am assuming overnight stays is the only variable you have missing data on). If the data is missing completely at random you can just delete the missing data, but its usually best to assume its missing at random unless you are sure its not and in any case you say you want the records regardless.

You next need to consider if overnight stays is an interval, ordinal, or categorical variable and how much data it is missing. And you have to determine if the pattern of missing data is montone or arbitrary. For monotone missing data with a continuous variable missing data you would use linear regression, predictive mean matching, or propensity scoring. For an interval variable with arbitrary missing data (which is more likely) you would use Markov Chain Monte Carlo.

Or so my books say. I have spent a lot of time reading material on missing data, but am hardly an expert

That scared me even hearing it.... What are you actually trying to do? With time series the simpler the better generally.Perhaps a (generalized) linear mixed effects model with a seasonal time series structure (is that even possible?) but I don't know if the missing data, the short time series, or the different interactions create a problem to build such a model.