We develop a method of synthetic frequency generation to construct an atomic clock with blackbody radiation (BBR) shift uncertainties below 10-19 at environmental conditions with a very low level of temperature control. The proposed method can be implemented for atoms and ions, which have two different clock transitions with frequencies ν1 and ν2 allowing to form a synthetic reference frequency νsyn = (ν1 - ϵν2)/(1 - ϵ), which is absent in the spectrum of the involved atoms or ions. Calibration coefficient ϵ can be chosen such that the temperature dependence of the BBR shift for the synthetic frequency νsyn has a local extremum at an arbitrary operating temperature T0. This leads to a weak sensitivity of BBR shift with respect to the temperature variations near operating temperature T0. As a specific example, the Yb+ ion is studied in detail, where the utilized optical clock transitions are of electric quadrupole (S → D) and octupole (S → F) type. In this case, temperature variations of ±7 K lead to BBR shift uncertainties of less than 10-19, showing the possibility to construct ultra-precise combined atomic clocks (including portable ones) without the use of cryogenic techniques.