My thought is it isn't the sh/pat but the bottle that may be the problem if she is pretty drowsy from that she might need that "full, I've just sucked milk" feeling to settle.
Can you try just putting her down drowsy from the milk without the sh/pat...does that work?
Also babies under 4-5months don't always know how to self sooth, so do need help.
If you can try & get her down into bed as awake as possible & only sh/pat when she is upset. If she is calm then let her work out if she can get to sleep, then you will probably have less wakings & if she does wake then maybe she is genuinely hungry.
HTH