Balb/c and C57BL/6 have very different immune response to parasites, specially malaria. For example, BALB/c mice are resistant against experimental cerebral malaria, while C57BL/6 is more susceptible (possibly due to excessive inflamation).
I like C57BL/6 because of disponibility of knockout mice (what can be very handy and enriching for your work). If I were you I would perform a comparative analyses between both strains and check if both Treg cells work properly (maybe C57BL/6 Treg cells don't work as they should and lead the exacerbated immune responde). You could also compare C57BL/6 (WT) with Foxp3sf (scurfy mice who have a deletion in the forkhead domain of Foxp3, fail to develop thymic-derived Foxp3+ regulatory T cells/ nTreg).
BALB/c should be the right choice for the study, it being the inbred strain and reported to generate a dominant TH2 response, required for clearing parasitic infections.