Yes. The combustion chamber only needs to receive the appropriate mix of fuel and oxidizer, it needn't know how it was put together.

Titan's atmosphere is about 97% inert nitrogen and 2.7% methane, which is below the flammability limit for this gas. So you would need a small distiller to separate and compress the methane, and a heater to mix it up with the oxygen. The heating part you can do on the cheap, using a radioisotope thermal generator. You can even go cheaper if your engine is not started and stopped frequently, providing the preheating energy using a condenser from excess heat that your engine naturally generates. In that case, you would need a battery/generator based heater only for starting until it gets warm to the point of self-sufficiency.

You want a "rich burn" to save on oxygen, so a FAR of 16:1; this means proportions of about 3.5 oxygen, 12.5 nitrogen, 1 methane. You need to enrich the methane proportion from 2.7% to 8% (1:12.5), i.e. about a factor of three. Then you add oxygen (1 L every 3.85 L of enriched Titan air).

This means that if we consider oxygen as the "fuel" ("what you have in your tank"), on Earth one engine cycle is 1/(16+1), on Titan it is 1/(1+3.85); your tankage efficiency is 4.85/17 = only 28% of whatever the same engine would have on Earth. For the same mileage, you need three times as much tanks.

My figures are probably a little off because I ran the calculations at STP, while Titan surface pressure is 1.45 atm. On the other hand, thermal cycle efficiency should be higher than Earth's since the ambient temperature is a good 200 K lower.

on the possibility of "perpetual motion" on Titan

Combustion engines on Earth take fuel (let's say methane CH4) and the oxygen from the air (O2) and convert them into H2O and CO2. In the process, some nitrogen N2 gets sucked in and oxidised to NOx, subtracting useful energy and producing harmful nitroxides; which is why even hydrogen engines are not "clean", unless they burn pure, costly oxygen instead of nitrogen-containing, free air.

The final equation is that the chemical energy of H2O and CO2 added together is less than the chemical energy of CH4 and O2. The difference can be converted, with some efficiency loss, into heat and movement.

On Titan we have for free both N2 and CH4. If we add O2, the possible final compounds comprise CO2 and H2O, which have little energy (and less than N2 and CH4 combines, so we can operate conversion at a gain).

Actually, it is because we need lower-energy compounds that we need to bring along oxygen tanks for the engine.

But if there was some compound that needed no oxygen, say hydrogen cyanide HCN or (CN)2 cyanogen, or cyanamide, or anything else, that also had less energy than N2 and CH4 combined, then we could run the conversion and siphon off the energy difference to power our engine for free.

The "trick" would be that the conversion would need to be not spontaneous (otherwise Titan would have exploded long since); our "engine" would for example heat the CH4/N2 atmosphere in the presence of an appropriate catalyzer, expending some energy to supply heat, until the magic happened, then the expended mixture would be expelled, extracting useful work and heat to further the process, and the cycle would start again.

We would also need for the reaction not to be so strong as to be self-perpetuating, otherwise as soon as we started the engine, again Titan's atmosphere would explode.

It all depends on the availability of such an end product (I have done some math and unfortunately (CN)2 and HCN have too much energy to be useful; it would actually cost us to convert CH4 and N2 into those) and a suitably compact process. But it could be a workable idea, and after all, we needn't name this miraculous end compound.

(In Campbell's Aarn Munro series, the Teff-El people wage war using an unobtainium "atmospheric catalyzer" that has the effect of burning the atmospheric nitrogen, rendering the target planet scorched and effectively airless)

The engine doesn't care what is the source of the mixture it is burning in its cylinders. As long as it burns, the engine is happy and running.

Therefore it is surely possible to have a tank of oxidizer providing the oxidizing agent to be mixed in the correct ratio with the combustible before reaching the combustion chamber and be burned.

If the oxidizer is stored in form of a liquid (i.e. liquid oxygen) you need the usual additional elements to evaporate the liquid preventing the cryogenic temperatures from negatively affecting your system. Maybe you can even use part of the waste heat for this scope.

You might also need additional protection on some engine parts from the pure oxidizing element, and don't forget that nitrogen in our atmosphere has a good role in mitigating the temperatures inside the combustion chamber. Burning pure mixture might overheat your engine if not properly taken into account during the design.