After the BPSK modulator, I present you the QPSK modulator.
How it works :
First, a square signal is generated at 4 time the frequency of the carrier. It's injected in a shift register.
The output of the shift register is 4 square signal dephased by 90°. They are the filtered (value of L and C are not optimized, 'cause, you know, 470uH is quite big).
Then the phase is selected using an analog mux/deux. OpAmp is optional
The whole circuit can be made using only two IC and a few passive components.
PS : simulation is damn slow on my phone