How would I connect an HC-06 bluetooth module to an SN75176A RS-485/422 transceiver?
I have deleted my other answer.
I was mistaken on the required interface, plus there were comments that the indicated chip was too expensive.

How about the MAX13487E/MAX13488E.  They are made for half-duplex two wire RS-485 networks and have auto direction control.  The cost is $2.65 at Mouser as of 05DEC19(Compared to $9+ for the MAX3162).  Which is give or take $1.25 more than the SN75176A.  The MAX13487E is slew limited(to cut down on EMI and is restricted to 500Kps) and the MAX13488E(allows up to 16Mps) is not.

Page 16 of the datasheet shows the auto direction circuit.  There is also a Design Solution/App Note that deals with subject matter.  It looks like at most it would require the MAX1348X, 6 resistors and 2 opto-couplers to interface it.  And the opto-couplers are only required if the HC-06 can not deal with it's RX and TX being pulled up.

Note, there is a prior question that compared the MAX485 with the MAX13487.