Let $E$ an elliptic curve over a finite field $\mathbb{F}_q$. Let $P_1, \dots, P_n \in E(\mathbb{F}_q) \backslash \{\mathcal{O}\} $ be $n$ distinct rational points of $E$. Suppose there exists a rational point $P \in E(\mathbb{F}_q)$ and $P \notin \{\mathcal{O}, P_1, \dots, P_n \}$.
My question: Is it possible to guarantee that $P_1 \oplus \cdots \oplus P_n \neq \alpha P$ for any nonzero $\alpha \in \mathbb{Z}$ (i.e., $P_1 \oplus \cdots \oplus P_n \notin \langle P \rangle-\{ \mathcal{O} \}$)?
If it is not possible, are there some conditions for that?
Context: I want to show that a divisor $\alpha P + \beta \mathcal{O} - P_1 - \cdots - P_n$ with $\alpha+\beta=n$ is not principal (or give some conditions for this). Using that a divisor $D$ with $\deg D=0$ on $E$ is principal iff $D = \mathcal{O}$ in the group law, I get to my question since $\alpha P \oplus \beta \mathcal{O} \oplus (-1) P_1 \oplus (-1) \cdots \oplus (-1) P_n = \alpha P \oplus (-1) P_1 \oplus (-1) \cdots \oplus (-1) P_n$
Thanks in advance.
Here is a counter-example.
It is a general fact that for a finite abelian group $A$ having odd torsion, $$\sum_{a\in A}a=0.$$ Now suppose $|E(\mathbb F_q)|>2$. Then let $P\ne O\in E(\mathbb F_q)$, and let $\{P_1,\dots,P_n\}=E(\mathbb F_q)\backslash\{O,P\}$. Then by the above $$P_1+\cdots+P_n=-P.$$