R, 90 bytes
\(x,y)Reduce(paste0,sapply(el(strsplit(x,"")),\(z)cwhmisc::int2B(grep(z, letters),y))[1,])
With R < 4.1, \ needs to be replaced with function resulting in a total of 104 bytes.
-5 bytes thanks to @pajonk.
Maël
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