abate is related to bate etymologically.
late Middle English: from Old French batre 'to beat' (see also
Middle English (in the legal sense): from Old French abatre 'to fell',
from a- (from Latin ad 'to, at') + batre 'to beat' (from Latin
battere, battuere 'to beat').
The word picture for bate is a hawk beating its wings against the air to fly. This word picture extends metaphorically to its noun definition: "foul mood".
Falconry (Of a hawk) beat the wings in agitation and flutter off the
the hawks bated and immediately the breeze got in their feathers
British informal (dated) An angry mood:
he got into a stinking bate
"Bate" obtains the meaning of "reducing the intensity of a force" as Shakespeare used it in Merchant of Venice. In each of the uses below, bate can reflect the meaning "subdue", although in some of them it could also reflect the original word picture of "beat" or "flutter".
"With bated breath and whispering humbleness,"
"His tedious measures with the unbated fire"
"These griefs and losses have so bated me,"
"And bid the main flood bate his usual height;"
By adding the prefix "a" (a phonetic variant of "ad") to the base of bate we get the picture of beating "at" something or someone.
Make (something) less intense:
nothing abated his crusading zeal
Again, abate obtains the meaning of "reducing the intensity of a harmful force" from the outcome of beating that harmful force back. In fact, "subdue" is the primary connotation of abate.
This is the clear meaning in Shakespeare's Merchant of Venice, when BASSANIO says,
"You would abate the strength of your displeasure."
So in the end, the common etymology creates common meaning across the board, although bate is rarely used now, except for the phrase "bated breath."