At last, I have finally managed to construct a device with a respectable signal quality. The circuit was completely redesigned, with no subcircuit left unchanged. Amongst the many changes are:

• The power supply was changed from single-supply to dual-supply (I'm currently using two 9V batteries). This allows to use high quality, low noise op-amps, and also reduces noise by moving supply currents away from the ground net.
• The oscillator is now implemented using the 74HCU04 and a 1 MHz ceramic oscillator.
• The supply for the oscillator is a low-noise LDO.
• The current excitation circuit was replaced by a JFET-based, cascoded and denoised current source, switched by a SPDT analog switch through a center-tapped transformer.
• The AM demodulator was completely removed! It turns out that the output signal is already present on the transformer's center tap. Therefore, the SPDT analog switch with the transformer serves both as an excitation circuit and an AM detector. As far as I know, the technique is novel - I haven't seen such a circuit in any patent or paper I have read on the subject.
• I'm using an ultralow noise op-amp (currently an AD8597) for the preamplifier.
• I have changed the lowpass filter topology to multiple feedback and added gain to the filter circuit. I currently have a gain of 100 in the preamplifier and gain of 10 in the active filter.
• The common-mode noise from the probes is additionally filtered using two methods: a common-mode choke in series with the probes and a center-tapped inductor with the center tap connected to ground. The technique is inspired by the common-mode filters used in Ethernet connectors. The filter significantly reduced pickup of external noise.

I have assembled the prototype on a perfboard, the required SMD components were mounted on adapters. Even though the prototype looks messy, it seems to work really well. I believe it will work even better when assembled on a well-routed PCB.