Searching based on lookup fields and adding fields to results from a lookup are two separate things, so you will need to do two steps to achieve this.
First, configure your lookup as an automatic lookup on your data. That'll allow you to leave off the explicitly | lookup command from your first search.
Second, run a search like this:
index=foo sourcetype=bar [inputlookup mylookup | fields month | rename month as date_month] | ...
That'll build an OR'd list of date_month filters from your lookup.
As an alternative, having the automatic lookup also allows you to write this kind of search:
index=foo sourcetype=bar shortname=jun | ...
That will be translated to date_month=june under the hood.
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