Deriving a general commutation relation for X,y.
Here y,b,a,x are the generators with relations
ax=x(a-α)  ay=y(a+α)
by=y(b-β)  bx=x(b+β)
xy=yx+(B-A)/ℏ
where A = (e^(-ℏ β a)), B = (e^(-ℏ α b)) and q = (e^(-α β ℏ))
AB = t is a central element. Since we want to expand in β,
we change variables to X = (Axt^(-1)) ℏ to get
Xy = q yX + 1-G where G = (A^2)(t^(-1)), so X G = (q^2) X G and G y = (q^2)y G.
also set α=1.

Below we verify the following fundamental commutation relation:

(X^m)(y^n) = (∑_(j=0)^(Min[m,n]))(q^((m-j)(n-j)))(([m]![n]!)/([m-j]![j]![n-j]!))(y^(n-j)) (   (∏_(i=1)^j)(1-(q^(m+n-j-i)) G) )(X^(m-j)) 
This formula should lead to a reasonable formula for commuting (e^X)(e^y)

(*γ /: (γ^(x_))/;x>1 :=0;*)
PBWRule={y->1,a->2,G->3,X->4};

B[U@a,U@G]=-(B[U@G,U@a]=0);
B[U@y,U@a]=-(B[U@a,U@y]= U[y]);
B[U@a,U@X]=-(B[U@X,U@a]= -U[X]);

B[U@G,U@X]=-(B[U@X,U@G]=((q^2)-1)U[G,X]);
B[U@y,U@G]=-(B[U@G,U@y]=((q^2)-1)U[y,G]);

B[U@y,U@X]=-(B[U@X,U@y]=(q-1) U[y, X]+U[]- U@G);

UU[l___,((x_)^(n_)),r___]:= UU[l,Sequence@@Table[x,{n}],r];
UU[l___,1,r___]:= UU[l,r];
UU[]=U[]; UU[l_,r___]:=U[l]**UU[r];
(Subscript[U, i_])[ℰ_] := ℰ /. {t->(Subscript[t, i]), u_U :> UU@@Replace[u, x_ :> (Subscript[x, i]),1]};

B[x_,x_]=0;
B[U[(Subscript[(x_), i_])],U[(Subscript[(y_), i_])]]:=(Subscript[U, i])[B[U@x,U@y]];
B[U[(Subscript[(x_), i_])],U[(Subscript[(y_), j_])]] /; i=!=j := 0;
B[x_,y_] := x**y-y**x;

x_⪯y_:=OrderedQ[{x,y}/.PBWRule]; x_≺y_:=!OrderedQ[{y,x}/.PBWRule]; 
Simp[ℰ_]:= Collect[ℰ, _U,Together];

Unprotect[NonCommutativeMultiply];
NonCommutativeMultiply[x_] := x;
0**_=_**0=0;
x_**U[]:=x; U[]**x_:=x;
(a_*x_U)**(b_*y_U) := If[a b === 0, 0, Simp[a b (x**y)]];
(a_*x_U)**y_ := Simp[a(x**y)]; x_**(a_*y_U) := Simp[a(x**y)];
(x_Plus)**y_ := (#**y)& /@ x; x_**(y_Plus) := (x**#)& /@ y;
U[x_]**U[y_] := (*U[x]**U[y] =*) If[x≺y, U[x,y], U[y,x]+B[U@x,U@y]];
U[x_] ** U[y1_,yy__] := (*U[x] ** U[y1,yy] =*)If[x⪯y1,U[x,y1,yy],(U@x**U@y1)**U@yy];
U[xx__,xn_]**U[yy__] := (*U[xx,xn]**U[yy] =*) U@xx**(U@xn**U@yy);

(*Gaussian quantum integer, factorial, binomial*)
QI[n_]:=((q^n)-1)/(q-1)
QF[n_]:=Product[QI[j],{j,1,n}]
QB[n_,k_]:=QF[n]/(QF[k]QF[n-k])

UU[X,y]//Simp
UU[(X^2),y]//Simp
UU[(X^3),y]//Simp

(*Testing the fundamental commutation relation*)
TestRel[m_,n_]:=
-UU[(X^m),(y^n)]+Sum[(q^((m-j)(n-j)))QB[m,j]QB[n,j]QF[j]UU[(y^(n-j))]**NonCommutativeMultiply@@Table[(U[]-(q^(m+n-i-j)) U[G]),{i,1,j}]**UU[(X^(m-j))]
,{j,0,Min[m,n]}]//Simp
TestRel[4,5]

Series[QF[4]/4!,{q,1,4}]

Series[QF[5]/5!,{q,1,5}]

Series[QB[4,2]/Binomial[4,2],{q,1,5}]