We performed a laser spectroscopic determination of the $2s$ hyperfine splitting (HFS) of Li-like ${}^{209}{\text{Bi}}^{80+}$ and repeated the measurement of the $1s$ HFS of H-like ${}^{209}{\text{Bi}}^{82+}$. Both ion species were subsequently stored in the Experimental Storage Ring at the GSI Helmholtzzentrum f\"ur Schwerionenforschung Darmstadt and cooled with an electron cooler at a velocity of $\ensuremath{\approx}0.71\phantom{\rule{0.16em}{0ex}}c$. Pulsed laser excitation of the $M1$ hyperfine transition was performed in anticollinear and collinear geometry for ${\text{Bi}}^{82+}$ and ${\text{Bi}}^{80+}$, respectively, and observed by fluorescence detection. We obtain $\ensuremath{\Delta}{E}^{(1s)}=5086.3(11)\phantom{\rule{0.16em}{0ex}}\mathrm{meV}$ for ${\text{Bi}}^{82+}$, different from the literature value, and $\ensuremath{\Delta}{E}^{(2s)}=797.50(18)\phantom{\rule{0.16em}{0ex}}\mathrm{meV}$ for ${\text{Bi}}^{80+}$. These values provide experimental evidence that a specific difference between the two splitting energies can be used to test QED calculations in the strongest static magnetic fields available in the laboratory independent of nuclear structure effects. The experimental result is in excellent agreement with the theoretical prediction and confirms the sum of the Dirac term and the relativistic interelectronic-interaction correction at a level of 0.5%, confirming the importance of accounting for the Breit interaction.