Generating points inside convexhullmesh

How could I generate these points

hexagon={{0, Sqrt[3]/2}, {1/2, Sqrt[3]}, {3/2, Sqrt[3]}, {2, Sqrt[3]/2}, {3/2,
   0}, {1/2, 0}}

following this rule

pts[x_, y_] := Flatten[Table[{{3 j, Sqrt[3] k}}, {j, 1, x}, {k, 0, y}], 2]];

but inside this region

ConvexHullMesh[{t1, t2, t1 + t2, {0, 0}}]

where

m = 1.;
n = m + 1;
a1 = {Sqrt[3]/2, -1/2}*Sqrt[3];
a2 = {Sqrt[3]/2, 1/2}*Sqrt[3];
t1 = m*a1 + n*a2;
t2 = (n + m)*a1 - m*a2;

I tried to use something like this

totalpoints=TranslationTransform[# - hexagon[[1]]][hexagon] & /@ 
  pts[somesize, somesize];

And then I cut inside the region doing this

pointsinsideregion = 
 Table[If[RegionMember[
    ConvexHullMesh[{t1, t2, t1 + t2, {0, 0}}], totalpoints[[u]]], 
   totalpoints[[u]], Nothing], {u, 1, totalpoints[Upp]}]

The problem is that I use more points that is necessary (in function pts[x,y] where somesize i get usually in terms of "m") to the calculation.

Maybe the solution is something using regionmember with the function pts[x,y]

.
.
.
Just to illustrate

eta = Graphics[{EdgeForm[Thickness[0.001]], PointSize[0.009], Blue, 
    Point /@ TranslationTransform[# - hexagon[[4]]][hexagon] & /@ 
     pts[5, 5]}];
Show[ConvexHullMesh[{t1, t2, t1 + t2, {0, 0}} + 4 m], eta, 
 Frame -> True, PlotRange -> Full]

m = 1.;
n = m + 1;
a1 = {Sqrt[3]/2, -1/2}*Sqrt[3];
a2 = {Sqrt[3]/2, 1/2}*Sqrt[3];
t1 = m*a1 + n*a2;
t2 = (n + m)*a1 - m*a2;
hexagon = CirclePoints[6];
pts[x_, y_] := # + {x, y} & /@ CirclePoints[6]
reg = ConvexHullMesh[{t1, t2, t1 + t2, {0, 0}}];
lattice = 
  DeleteDuplicates@
   Flatten[Table[
     pts[3 x/2, Sqrt[3] y + 3 Sqrt[3]/2 x], {x, -1, 5}, {y, -10, 0}], 
    2];
inside = Select[lattice, RegionMember[reg, #] &];
Graphics[{Gray, reg, Red,
  Point[lattice],
  Blue, Point[inside]
  }, Frame -> True, PlotRange -> {{-1, 8}, {-5, 4}}]