ValueError from statsmodels ExponentialSmoothing
The error is raised from lead - lag; in initial_values, these are set as y[m:2m] and y[:m] respectively,  where m is the seasonality length (52 in your case). So lead is only getting 22 values, and hence the size mismatch complaint.
So yes, for this to work directly, you'd need at least 104 datapoints.  See the source, and docpage.
(N.B.  This is for version 0.10.  Some changes have been made in v0.12, in which the initialization takes more options.  But OP's context is version-specific, so I'll leave it at that.)
But, you can bypass this bit of the code by passing your own initial_slope to fit*.  I might suggest mimicking statsmodels's approach, but limiting the year-over-year slopes to the range you have available: (y[52:74] - y[:22]).mean() / 52
* this is one change in v0.12; these parameters are passed on model instantiation, not fit.  See the docs if you want to do this in a newer version.