The impact of missing data (including for covariates) is a growing concern for meta-analytic methodologists (see Cooper & Hedges, 2009; Pigott, 2012).
In terms of how to proceed, analytically, Mike Cheung's metaSEM() package (see Cheung, 2015a) for R is one of the only options that I know of that has features which explicitly deal with the missing covariate data problem. His documentation here details how his
meta3x()function uses full-information maximum likelihood (FIML) with a structural equation modelling "trick" to address covariate missingness. It looks slick; the only complication (vs. the package's other functions) is that users must specify at what level their moderator variables (and auxiliary variables, if any) are located. His book (Cheung, 2015b) also provides a nice overview of the missing data issue in meta-analysis, including a more thorough description of the FIML process his package uses.
I'm not aware of anyone commandeering multiple imputation (yet) to handle missingness in meta-analysis, as you are attempting here. While it seems reasonable, in principle, to expect MI to serve this purpose, I wonder/worry if/how your MI procedure is weighting each study while drawing information from the covariance of auxiliaries/effect sizes and your non-missing moderator values, to fill in the missing values.
Regarding your concerns about "playing tennis without the net", I think there are a few reasons why this shouldn't trouble you too deeply. For one, MI/FIML will not just take the association between your observed moderator variables and effect sizes/auxiliary variables into account when "filling in" missing values; it will also take the uncertainty/imprecision of that association into account, and the less complete data you have, the more modest/uncertain those "guesses" for MI will be.
Secondly, there's pretty clear evidence at this point that under conditions of Missing Completely At Random (MCAR) and Missing At Random (MAR), that MI and FIML trounce deletion methods and simple substitution methods, in terms of bias and statistical power (see Enders, 2010, for a review). True, more bias creeps in and recovery (and therefore statistical power) is worse when missingness is not random (MNAR), but this is true for all missing data treatments, and the amount of bias introduced is still less for FIML compared to these other methods.
Finally, consider an example of missing data "circularity": imagine you have very good reason to trust that there is a detectable and appreciable difference in height between men and women. Then you encounter a data set in which there is some missing height data, but you know the birth-sex of the individuals in question. Is it playing tennis without the net to take into account the known association between sex and height? Or is maximizing your use of previous knowledge to make more accurate and efficient inferences? Surely there is a line at some point and an accompanying grey-zone where the difference between the two uses becomes hazy. But I don't think your circularity concern is as clear cut as it first seems, because there will be cases where, to not use advanced missing data treatment techniques, would essentially require you to hurt the power/accuracy of your investigation for fear of benefiting from previous knowledge.
Cheung, M. W. L. (2015a). metaSEM: An R package for meta-analysis using structural equation modeling. Frontiers in Psychology, 5, 1521.
Cheung, M. W. L. (2015b). Meta-analysis: A structural equation modeling approach. Chichester: Wiley.
Cooper, H., & Hedges, L. V. (2009). Potentials and limitations. In H. Cooper, L. V. Hedges, & J. C. Valentine (Eds.), The handbook of research synthesis and meta-analysis (pp. 561-572). New York, NY: Russell Sage.
Enders, C. K. (2010). Applied missing data analysis. New York, NY: Guilford Press.
Pigott, T.D. (2012) Advances in Meta-Analysis. New York, NY: Springer.