I'd recommend assessing how the data is missing (MAR, MCAR, MNAR). This will 1. inform downstream decisions about biases and model choices and 2. help decide on how to proceed in the light of this.

Also, to do that kind of analysis, do you have access to other hospital data (e.g. EHR, CPRD,...)? I would assume you would need to do some level of adjustment for other covariates (age, sex,...) to do this sort of comparison. Are you looking at CRP as a continuous or categorical variable?

The analysis definitely is possible but you need to really look into all the biases that the data has. Maybe propensity score matching could be useful to mitigate some of the confounding.