Strength of Hash obtained by XOR of parts of SHA3
Truncating a hash does reduce the collision resistance. Mixing in an independent string does not.

$$
\text{random thing} \oplus \text{independent thing} \to \text{random thing}\\
\text{random thing} \oplus \text{independent random thing} \to \text{random thing}
$$

Truncating hashes is common and the reduction is relative to the bits lost. The truncated $\text{sha3}_{384[320]}$ should be stronger than $\text{sha2}_{256}$ and is weaker than $\text{sha3}_{384}$.

I recommend that you do not try to mix in the truncated $64$-bits.