No farang or tourist bars near the MFA, but there is a restaurant downstairs, the MFA is in part of Bangkok with lots of government buildings, not much in the way of hotels there.
The British embassy is next to the BTS Ploen Chit station, so hotels/guesthouses from soi 21 Asoke to soi 4 Nana to siam square will be close to the BTS and you also will need the BTS to get to Mo chit and then a taxi to the MFA.
Try trip advisor for the accommodation.
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