Possibly not the answer you are looking for but if your filter contains over 1,000 values then I'd suggest this isn't best surfaced to users as a drop down list - have you considered giving users a wildcard filter instead?
Can you confirm how the same filter performs when not embedded? I'm curious if this is related to your embed code or if this is consistent in both instances. You mention being on version 10.1 of Desktop, this could potentially be an issue that has been resolved in a newer version of Tableau.
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