Consider the increment of Brownian motion. It is of infinite variation, so the sum $$\sum_i^{n-1} | B(t_{i+1})- B(t_i) |$$ for all possible partitions of the time interval $[0,T]$ is not bounded. Now, consider that we add a variable $a_t >0$, such that we have $$\sum_i^{n-1} a_t | B(t_{i+1})- B(t_i) |$$ where $\sum_i a_{t_i}^2 < \infty$ but $\sum_i a_{t_i} = \infty$.
Which conditions should the variable $a$ satisfy to make this sum of increments bounded?
Indeed as mentioned in the comments, if you allow for random choices, then you can just take $a_{t_{i},t_{i+1}}=| B(t_{i+1})- B(t_i) |$.
If you want deterministic choice, then you can use the modulus results for BM from the Peres-Morters book on Brownian motion lemma 1.16
So on top of the constraints you mentioned we also require
$$ \sum a_{t_{i},t_{i+1}}\sqrt{(t_{i+1}-t_{i})\log\frac{1}{t_{i+1}-t_{i}}}<\infty$$
so that
$$\sum a_{t_{i},t_{i+1}}| B(t_{i+1})- B(t_i) |<\infty.$$