Solution:

Let a lateral move refer to one which is either up or right. Then the lateral moves are the only ones which increase Kelvin's sum of coordinates by $1$, while all other moves do not change the sum, so Kelvin must make $6$ of them, one to increase this sum from $i$ to $i + 1$ for each $i \in [0, 5]$.

We claim there exists a unique path corresponding to each set of $6$ lateral moves chosen in this way. Indeed, the diagonal moves allow Kelvin to get from any point on $x + y = i$ to any second point on $x + y = i$ in exactly one way.

Observe that when $i \leq 2$, the number of lateral moves that increase the sum of coordinates from $i$ to $i + 1$ is $2i$, as is the number that increase it from $5 - i$ to $6 - i$. Thus the answer is $2 \cdot 4 \cdot 6 \cdot 6 \cdot 4 \cdot 2 = \lfloor 2304 \rfloor$.