Derivation condition to obtain the correct asymptotic link

Suppose that $ X sim text {Bin} (n, theta) $. By the local limit theorem (Theorem 7 here) for discrete random variables,
$$
P (X = t) = frac {1} { sqrt {2 pi n theta (1- theta)}} exp left (- frac {(tn theta) ^ 2} {2n theta (1- theta)} right) + o (n ^ {- 1/2})
$$

for all $ n geq 1 $ and uniformly in the integers $ t $.

I'm interested in the form. $ t = n theta + sqrt {2 theta (1- theta) n log m + O (1)} $ and the relationship between $ n $ Y $ m $. In particular, $ m >> n >> 1 $. For example, in my application, $ m approx. $ 20000 Y $ n approx. $ 200 it seems to work well

I am interested in finding a theoretical relationship between $ n $ Y $ m $ such amount $ mP (X = t) = O (1) $. Now if the $ o (n ^ {- 1/2}) $ the rest was not there, so I can reason the following,

begin {align *}
mP (X = t) & = O (mn ^ {- 1/2} exp (-O ( log m))) \
& = O left ( exp left ( log m- log n ^ {1/2} -O ( log m) right) right) \
& = O left ( exp left ( log m ^ {r} – log n ^ {1/2} right) right) text {for some constant $ r> 0 $} \
& = O left ( frac {m ^ {r}} {n ^ {1/2}} right)
end {align *}

This suggests that $ m leq n ^ gamma $, where $ gamma = frac {1} {2r}> 0 $ for $ r> 0 $ It can be a reasonable relationship.

How do I manage that rest? $ o (n ^ {- 1/2}) $? Intuitively, $ m $ should grow much faster than $ n $.