Note that additive overlapped windows of pure sinewaves of different phases will not produce a pure sinewave. The difference may include something like a low frequency beat note, which might explain the low frequency content in the 2nd spectrum.

Splitting the magnitude of your between bin frequency between the two (or more) nearest bins using Sinc interpolation might reduce your artifact.

Note that additive overlapped windows of pure sinewaves of different phases will not produce a pure sinewave. The difference may include something like a low frequency beat note, which might explain the low frequency content in the 2nd spectrum. Modulation side-bands might explain the nearby higher peak. See trigonometric sum and difference formulas.

Splitting the magnitude of your between bin frequency between the two (or more) nearest bins using Sinc interpolation might help reduce your artifacts somewhat, depending on how you do your delta phase correction/interpolation between windows.