How to add a stable DC voltage and a DC pulse train together

A "normal" relay (mechanical or solid state), as mentioned in the question, can definitely be used for switching both AC and DC supplies. What you need to look out for is the different contact ratings that a mechanical relay or switch will have for AC voltage and current, DC voltage and current, and reactive AC voltage and current. This last is typically applicable to inductive / electromagnetic coil loads, and should not be of any relevance for the "tiny capacitance" of your electrode. This answer provides additional detail.

Something like the RR2KP series power relays may suit the requirements:

However, for frequent switching, mechanical relays are not recommended both due to mechanical noise, and due to short operational life in terms of number of actuations.

Preferably, a Solid State Relay (SSR) rated for the DC and AC voltages and currents envisaged, can be used as well. SSRs are a lot more forgiving about difference between AC and DC ratings. Rating for DC voltage, and AC peak voltage, are often the same on SSRs.

Also, SSRs do not typically have contact resistance increasing over time, as there are no actual metal contacts that can degrade through pitting, oxidation et cetera.

For example, the Omron G3VM-352C/F SSR may meet the requirements:

SSRs are typically rated for many more actuations than a mechanical relay. This would fit in with the requirement for switching rather frequently.

SSRs are available in really tiny surface mounted packages, so the size constraint mentioned is also addressed.

SSRs are typically much more expensive than typical mechanical relays, though.

If the various supply voltages mentioned are fed in through a single connection ("hot" and neutral / ground), then a simple SPST relay (mechanical or SSR) can be used, if it meets the following criteria:

• Actuation voltage and current are within the capability of your microcontroller circuit, or an added driving element, e.g. MOSFET switch, is incorporated in the circuit.
• Both the DC and AC supplies are within the respective ratings of the relay contacts.
• The relay datasheet actually specified both AC and DC operation / ratings. Don't assume a relay is good for DC if not specified, for instance.

If, however, (and this isn't clear to me from the question) the DC and AC supplies are fed in on different connections, then a DPST (Double Pole Single Throw) relay would be needed... though some high voltage industrial applications use two separate relays for such purpose, if only to provide greater physical isolation between the two lines and prevent inter-circuit arcing during turn-off.

I would use a shift register - use a constant to initialise it to your desired pulse train when reset is asserted and then just clock it out. Your top-level entity will only need three pins, clock, reset and the output.