For a 20 kW engine with a 20 L tank and a bore of about 10 cm, the tank's temperature should be about 300 K (isothermal) to 400 K (adiabatic, more practically), says p. 11 of an analysis by Sandia Labs, "Optimizing the Scavenging System for a Two-Stroke Cycle, Free Piston Engine for
High Efficiency and Low Emissions: A Computational Approach," by Goldsborough and Van Blarigan, SAE Trans. 112(3): J. Engines, pp. 1-20.
400 K is 125 ºC, comfortably cool.
Newcastle U. investigated (unpaywalled) cylinder wall temperatures from 370 K to 435 K (p. 383, top) and a modified "low heat rejection design" with wall temperatures up to "50 per cent" higher (p. 384), thus at most 650 K.
(Mikalsen & Roskilly, "The fuel efficiency and exhaust gas emissions of a low heat rejection free-piston diesel engine," Proc. IMechE (A): J. Power and Energy 223:379-384.)
650 K is 375 ºC, still cool enough for non-exotic piston materials, despite the in-cylinder gas temperature ranging from 400 K to 1800 K through a piston stroke (p. 383, Fig. 4).
But Fig. 8 shows that the steady-state exhaust gas temperature is 1100 K to 1140 K (870 ºC), so the turbine itself must be made of sterner stuff.