Using quaternions instead of 4x4 matrices for transformations

matrices quaternions transformation

To apply a subsequent rotation corresponding to $q_1$ and translation corresponding to $T_1$ to your $v'$, you'd just do it in terms of quaternion algebra: you'd get

$v''=q_1v'q_1^{-1}+T_1=q_1(qvq^{-1}+T)q_1^{-1}+T_1=(q_1q)v(q_1q)^{-1}+q_1Tq_1^{-1}+T_1$

Thus, as you say, it is possible to do the computations on the transformations cumulatively, and then apply it to the data, to avoid cumulative distortion in the data.

(edited to correct original typo in formula)

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