Gate CD45+, CD11b +, CD11c-/Ly6C + population. From this Ly6C+, gate out Ly6G + neutrophils. Then the final Ly6C hi are classical and Ly6C lo are non-classical monocytes

The most classical way to stain would be to add CD115/MCSFR to your panel (requires constant EDTA and cold temperatures to prevent cleavage). CD115+ CD11b+ cells would be monocytes and Ly6C could be used to distinguish classical and nonclassical. For your current markers, while nonclassical monocytes have expression of CCR2-RFP in reporter mice (though less than classical monocytes), I haven't used the anti-CCR2 antibody and am not sure how reliable it is. When an antibody isn't staining well and a reporter mouse exists, it may be that the antibodies aren't great. F4/80 won't identify blood monocytes. CD43 apparently is on nonclassical monocytes (PMID: 25373902, used after CD115 staining. I believe any flow cytometry Misharin publishes), but I don't think it is specific. I would be concerned that any gating strategy you devise for nonclassical monocytes using the current markers would capture other myeloid cells (non-neutrophil granulocytes). You could compare the side scatter profile of your classical monocyte gate, nonclassical monocyte gate, and Ly6G+ neutrophil gate. If the granularity of the "nonclassical monocytes" looks like neutrophils, that would be a clear sign the gating is not working. CX3CR1 is another marker of nonclassical monocytes in the blood, but I've always used reporter mice. I don't think the antibody is always reliable. I would recommend adding CD115 if possible.

Something else to consider is that in settings of inflammation and/or emergency hematopoiesis is that neutrophil progenitors (which are Ly6C+) can express less Ly6G. Without CD115 staining, these could be misidentified as nonclassical monocytes