There is no strategy that is guaranteed to ever win, thanks to the BOOM rule.
If Alice picks $N = a^{124}\times b$, with $a$ and $b$ distinct primes, then $N$ has 250 factors. Bob's challenge is to determine $a$ and $b$.
The problem is that if he determines the order of the factors before determing both $a$ and $b$, then the BOOM rule can prevent him from ever learning $b$.
For example, if he asks for $d_{50}$ and gets back $3^{25}$, then he knows that $N = 3^{124}\times b$, and that $b = 5$ or $b = 7$. He also knows that all the even-indexed factors (up to 248) are $d_{2i} = 3^i$ and odd-indexed factors after $1$ are $d_{2i+3} = 3^i\times b$. And of course, he knows $N = d_{250} = 3^{124}\times b$.
If he ever asks about any odd-indexed factor (other than 1), he gets a BOOM, because he knows the corresponding other factor. For example, he knows that $d_{248} = 3^{124}$, so if he asks about $d_3$, he gets a BOOM. Thus, he can never determine whether $5$ or $7$ is the other factor.
This extends generally to the case where $a^k < b < a^{k+1}$. If he asks about a factor (where $i \leq k$, $j \geq 0$), then the answer will be:
$$ d_{i} = a^{i-1}$$$$ d_{250-i} = a^{124-i}\times b$$$$ d_{k+2j+1} = a^{k+j}$$$$ d_{k+2j+2} = a^j \times b$$
If he somehow determines $k$ before he determines $b$, then he has determined half the factors. He can't ask about the other half, because of the BOOM rule. Specifically, he can't ask about any factor containing $b$, so he can't ever determine $N$.
The only way to be sure he doesn't determine $k$ before $b$ is to start at $d_2$ and ask about each one going up. Unfortunately, if he gets to $d_{125} = a^{124}$, then he has lost because it must be that $b > a^{124}$ and all the remaining factors are just $b$ times the ones he's already gotten.
Thus, there is no strategy that guarantees a win in any number of turns.
Update:Alice could have picked $N = a^{249}$ instead, as that also has $250$ factors. Bob would not know which after asking about $d_2$ up to $d_{125}$. Unfortunately, he still can't ask about any factor past number $126$, so he's still sunk.