How to obtain the in-phase carrier for synchronous detection of an AM signal?
You can't demodulate full and proper DSBSC (double sideband, suppressed carrier) by trying to filter the carrier frequency but, you should be able to do this on your design because you are adding a dc offset to your modulation and this means that there will be carrier signal present that is "gettable" with filtering. 

You should consider using a resonant tank circuit to achieve this because, your modulation method may over-modulate and reverse the phase of the carrier in the composite signal. The resonant tank will keep the coherent carrier present in the detector despite it reversing on parts of the composite signal. However, if you over-modulate too much this method produces rapidly diminishing results and that leaves you with fairly complex methods such as the Costas detector or a squaring detector, both shown below: -