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The bulk abundances of exoplanetesimals can be measured when they are accreted by white dwarfs. Recently, lithium from the accretion of exoplanetesimals was detected in relatively high levels in multiple white dwarfs. There are presently three proposed hypotheses to explain the detection of excess lithium in white dwarf photospheres: Big Bang and Galactic nucleosynthesis, continental crust, and an exomoon formed from spalled ring material. We present new observations of three previously known lithium-polluted white dwarfs (WD J1824+1213, WD J2317+1830, and LHS 2534), and one with metal pollution without lithium (SDSS J1636+1619). We also present atmospheric model fits to these white dwarfs. We then evaluate the abundances of these white dwarfs and two additional lithium-polluted white dwarfs that were previously fit using the same atmospheric models (WD J1644-0449 and SDSS J1330+6435) in the context of the three extant hypotheses for explaining lithium excesses in polluted white dwarfs. We find Big Bang and Galactic nucleosynthesis to be the most plausible explanation of the abundances in WD J1644-0449, WD J1824+1213, and WD J2317+1830. SDSS J1330+6435 will require stricter abundances to determine its planetesimal's origins, and LHS 2534, as presently modeled, defies all three hypotheses. We find the accretion of an exomoon formed from spalled ring material to be highly unlikely to be the explanation of the lithium excess in any of these cases. [ABSTRACT FROM AUTHOR]