The following is (now apparently disproven) speculation based on some very basic radio theory rather than any particular knowledge of Sputnik 1. It should be disregarded if someone with knowledge of the actual circuits in use has better information. I did some web searching but did not find any descriptions of Sputnik 1 that provided good evidence for or against the below theory about its construction.
If we suppose that 40 MHz is not chosen for reception reasons, then the obvious reason to choose 40 MHz is that it is the second harmonic of 20 MHz. An oscillator generally produces some amount of signal at harmonics of the original frequency (small or large depending on its design), and certain RF circuit elements such as stubs (and, with some complications, antennas) also work with harmonics as with the fundamental. Therefore, components could be shared between the 40 MHz and 20 MHz systems, thus saving weight and complexity.
[Further information indicates that this was not the case — the 20 and 40 Mhz transmitters were separate, and considered redundant systems to each other. There still might have been some design reason to use an approximate multiple, but I can't say what exactly they might be.]
Signals on the first frequency were transmitted in 0.3 s pulses … with pauses of the same duration filled by pulses on the second frequency. [Wikipedia]
While this alternation could be purely to even out the power draw, it also suggests that there might not be two independent transmitters, but one transmitter which either produced both signals at once in the oscillator, followed by a switchable filter to select one of them to send to the power amplifier or antennas, or the oscillator itself could be modulated to switch between those frequencies.
Sputnik is said in Wikipedia and a citation behind paywall to have transmitted on "20.005 and 40.002 MHz". Of course, 20.005 × 2 = 40.010, not 40.002. But this does not mean harmonics are not applicable; under the theory that the second harmonic was being used to generate the 40 MHz signal, it is reasonably plausible that when the transmitter switched between the 20 MHz and 40 MHz conditions, secondary effects (changes in loading of circuits, etc) caused a slight frequency shift; or, if the oscillator is modulated, it's merely error or a free choice in the designed tuning, and the requirement is only that the second frequency be approximately a multiple of the first.
I don't know what the actual components of a one-watt tube-based transmitter of the time would be, so I don't know how plausible either of these hypotheses is. [Plausible, perhaps, but apparently false.]