This is an interesting problem. Some remarks:
fm[x_, m_, s_] := Exp[-(x - m)^2/(2 s^2)]/s
fp[x_, m_, s_] := Exp[-(x + m)^2/(2 s^2)]/s
f2[x_, m_, s_] := Exp[-x^2/(2 (m^2 + s^2))]/Sqrt[m^2 + s^2]
Table[Plot[{fm[x, i, j], fp[x, i, j], f2[x, i, j]},
{x, -10, 10}, PlotRange -> All],
{i, 1, 3}, {j, 1, 3}]
ff[x_, m_, s_] := fm[x, m, s] + fp[x, m, s] - 2 f2[x, m, s]
Manipulate[
Plot[{ff[x, i, j], Abs[ff[x, i, j]]},
{x, -5, 5}, PlotRange -> {-1, 1}],
{i, 1, 3}, {j, 1, 3}]
Integrate[Abs[ff[x, 1, 1]], {x, -Infinity, Infinity}] // N
NIntegrate[Abs[ff[x, 1, 1]], {x, -100, 100}]
Integrate[Abs[ff[x, 2, 1]], {x, -Infinity, Infinity}] // N
NIntegrate[Abs[ff[x, 2, 1]], {x, -50, 50}]
vtab = Table[
{i, j, NIntegrate[Abs[ff[x, i, j]], {x, -50, 50}]},
{i, 1, 3, .1}, {j, 1, 3, .1}];
ListPlot3D[Flatten[vtab, 1]]
tt = Table[{j, NIntegrate[Abs[ff[x, j, 1]], {x, -50, 50}]}, {j, 2, 20}];
ListLinePlot[tt]
It may be that the integral is bounded.