You can change the null hypothesis, that you have a fair coin with $p=0.5$ for flipping heads, into an equivalent null hypothesis that you have a fair coin with $p=0.5^5=0.0315$ for flipping five heads in a series of five flips.
If you are making 3 of such series then the distribution under the null hypothesis is
A problem here is to define what observation is considered as a rare event. With the likelihood ratio test one would consider the observations with the lowest likelihood ratio, which are
$$LH = \begin{cases}
\frac{0.90915}{1} & \approx & 0.90915 & \quad \text{if $X=0$}\\
\frac{0.08798}{0.4444} & \approx & 0.19769 & \quad \text{if $X=1$}\\
\frac{0.00284}{0.4444} & \approx & 0.00639& \quad \text{if $X=2$}\\
\frac{0.00003}{1} & \approx & 0.00003 & \quad \text{if $X=3$}
\end{cases}$$
these denominators are the probabilities if the alternative hypothesis equals the maximum likelihood estimate.
The p-value is the probability of the observed likelihood ratio or higher and is the sum of the probabilities for $X=1, X=2,X=3$ and as you calculated $\text{p-value} \approx 0.09085$.
Note, the power of your test can ne very bad for specific values of the alternative hypothesis. One could say that your observation $X=1$ is rare given the null hypothesis, however, it will be much more rare when the null hypothesis is wrong and $p<0.00335$. So the power of your experiment is very bad and when $p<0.00335$ you will be unlikely to reject the null hypothesis.
A likelihood function gives a better view of the result:
Here you see that the fair coin at $p=0.00325$ is not very well supported. Although the same would be true for other values of $p$, the experiment is not very strong.
