Your method works in one notebook session, but I get different results in different sessions.
In[1]:= genRands[{nRands_, randSeed_, randMethod_}, lbubs_?MatrixQ] :=
Block[{n, s, t, r, randMethodUsed}, n = Length[lbubs];
s = Table[{0, 1}, n];
t = RescalingTransform[s, lbubs];
randMethodUsed = If[randMethod == "", "ExtendedCA", randMethod];
SeedRandom[Method -> randMethodUsed];
r = BlockRandom[SeedRandom[Method -> randMethodUsed];
RandomReal[1, {nRands, n}], RandomSeeding -> randSeed];
t /@ r]
In[2]:= genRands[{5, 0, "Congruential"}, {{0, 1}, {1, 2}}]
Out[2]= {{0.658186, 1.33685}, {0.750221, 1.05406}, {0.554339, 1.60156}, {0.333813,
1.1567}, {0.306864, 1.94476}}
close notebook and restart genRands
In[2]:= genRands[{5, 0, "Congruential"}, {{0, 1}, {1, 2}}]
Out[2]= {{0.549267, 1.18874}, {0.813163, 1.91422}, {0.564126, 1.3203}, {0.337698,
1.46146}, {0.568895, 1.14699}}