POTS DAA circuit explanation
I'm not going to pretend to understand many aspects of how that circuit operates but I found United States Patent 6771696 that was issued to Silicon Laboratories that contains similar line-isolation circuitry. It also uses the same nonclemature which makes it easier to follow.

To answer the main part of question it appears the IC must have the line driver built-in. In the patent the DCT (DC termination) pin connects to the phone line via an attentuation resistor Rc. That is R1 in the above circuit so I'd try a value of say 820 ohms for a start and see what difference it makes. It sounds like you're connecting to privately owned equipment so won't get hassled by the phone company if you modulate it more than the usual phone standards.

From the patent and the datasheet block diagram QB, QE and QE2 are shown as part of the off-hook circuitry.  In the patent QB is shown with an internal MOSFET that goes to ground so I'd say that Q4 is being used to take the line off-hook and Q5 is used for off-hook detection.

It's outside the scope of the patent and I couldn't find a good explanation for DCT2 and the associated Q2 and Q3. Judging from the circuit my guess would be it's disconnecting parts of the circuit while off-hook to reduce loading or some other compliance issue. But it's hard to know without a description / diagram of how the chip operates internally which I couldn't find anywhere.