In p.415 of volume 3 of Gauss's werke one can find the following remark of Gauss:
[Later note]: I. $$\alpha+\delta+\gamma = \pi [=\varpi]$$ set $$\mathbb{sinlemn}(\alpha) = \mathbb{tang} (a), \mathbb{coslemn}(\alpha)=\mathbb{cos}(A)$$ $$\mathbb{sinlemn}(\delta) = \mathbb{tang} (b), \mathbb{coslemn}(\delta)=\mathbb{cos}(B)$$ $$\mathbb{sinlemn}(\gamma) = \mathbb{tang} (c), \mathbb{coslemn}(\gamma)=\mathbb{cos}(C)$$ then $a,b,c,A,B,C$ are sides and angles of a spherical triangle, and also: $$\frac{\mathbb{sin}A}{\mathbb{sin}a}=\frac{\mathbb{sin}B}{\mathbb{sin}b}=\frac{\mathbb{sin}C}{\mathbb{sin}c}=\sqrt{2}$$ II.$$\alpha+\delta+\gamma = \frac{\pi}{2} [=\frac{1}{2}\varpi]$$ set $$\mathbb{sinlemn}(\alpha) = \mathbb{cos} (a), \mathbb{coslemn}(\alpha)=-\mathbb{tang}(A)$$ $$\mathbb{sinlemn}(\delta) = \mathbb{cos} (b), \mathbb{coslemn}(\delta)=-\mathbb{tang}(B)$$ $$\mathbb{sinlemn}(\gamma) = \mathbb{cos} (c), \mathbb{coslemn}(\gamma)=-\mathbb{tang}(C)$$ then $a,b,c,A,B,C$ are sides and angles of a spherical triangle, and also: $$\frac{\mathbb{sin}A}{\mathbb{sin}a}=\frac{\mathbb{sin}B}{\mathbb{sin}b}=\frac{\mathbb{sin}C}{\mathbb{sin}c}=\sqrt{\frac{1}{2}}$$
So this note apparently establishes a link between lemniscate elliptic functions and spherical triangles. Despite having no idea what was Gauss's "hidden" intension in this note (why lemniscatic functions, which are related to the lemniscate curve, should have any connection to spherical geometry?), I noticed that it is easy to prove some of its statements - the statemets that resemble the spherical sine law.
To prove that $\frac{\mathbb{sin}A}{\mathbb{sin}a}=\frac{\mathbb{sin}B}{\mathbb{sin}b}=\frac{\mathbb{sin}C}{\mathbb{sin}c}=\sqrt{2}$ (in [I]), use the identities $\mathbb{coslemn}^2(x) + \mathbb{sinlemn}^2(x) + \mathbb{coslemn}^2(x)\mathbb{sinlemn}^2(x)=1$ and $\mathbb{sin}x = \frac{\mathbb{tan}x}{\sqrt{1+\mathbb{tan}^2x}}$, and you will get:
$$\frac{sin A}{sin a} = \frac{\sqrt{1+tan^2 a}\sqrt{1-cos^2A}}{tan a} = \frac{\sqrt{(1+\mathbb{sinlemn}^2\alpha)(\mathbb{sinlemn}^2(\alpha) + \mathbb{coslemn}^2(\alpha)\mathbb{sinlemn}^2(\alpha))}}{\mathbb{sinlemn} \alpha} = \sqrt{(1+\mathbb{sinlemn}^2\alpha)(1+\mathbb{coslemn}^2(\alpha))} = \sqrt{1+\mathbb{coslemn}^2(\alpha) + \mathbb{sinlemn}^2(\alpha) + \mathbb{coslemn}^2(\alpha)\mathbb{sinlemn}^2(\alpha)}=\sqrt{2}$$
and the same applies for angles $\delta,\gamma$. Therefore, the substitutions Gauss made here imply that ratios of the form $\frac{sin A}{sin a}$ do not depend on the size of the relevant angles, and hence have no connection to the condition $\alpha +\beta+\gamma = \pi$. This condition is apparently related to the possibilty of $a,b,c,A,B,C$ being realized as the sides and angles of a spherical triangle.
So how to prove from this condition that $a,b,c,A,B,C$ are the sides and angles of a spherical triangle on the unit sphere? an answer that will explain why there is such a connection between spherical geometry and lemniscate elliptic functions will be much appreciated!
After some intense "detective" work, I found the Dissertation "APPLICATIONS OF ELLIPTIC FUNCTIONS IN CLASSICAL AND ALGEBRAIC GEOMETRY" by Jamie Snape which is available freely on the web. Chapter 9 of this Dissertation deals with applications of elliptic functions to spherical trigonometry, and on section 9.2 (p. 60) appears a theorem which seems to be very close (or perhaps identical?) to Gauss's note. It defines the elliptic measures $u,v,w$ of the angles $\alpha,\beta,\gamma$ of a spherical triangle to be:
$$\mathbb{sn} (u) = \mathbb{sin} \alpha, \mathbb{sn} (v) = \mathbb{sin}\beta, \mathbb{sn} (w) = \mathbb{sin}\gamma$$
where each of the $\mathbb{sn}$ is one of Jacobi elliptic function (lemniscate functions are a special case of Jacobi elliptic functions), and the modulus $k$ of each is defined as the constant in the spherical law of sines, $k = \frac{\mathbb{sin} a}{\mathbb{sin}A}=\frac{\mathbb{sinb}}{\mathbb{sin}B}=\frac{\mathbb{sinc}}{\mathbb{sin}C}$.
Then the following theorem is proved algebricaly:
Here $2K = 2\int_0^1\frac{dx}{\sqrt{1-x^4}} = \varpi$, so this theorem appears to be related to the first result in Gauss's note. In addition, since $\mathbb{sn}^2(x)+\mathbb{cn}^2(x) = 1$, we get $\mathbb{cn}(u) = \mathbb{cos}\alpha$, and since the modulus $k$ is defined to be $\frac{1}{\sqrt{2}}$ (I showed it in my posted question) we get that $\mathbb{cn}(u)=\mathbb{coslemn}(u)$ (Jacobi elliptic functions become lemniscate functions when $k=\frac{1}{\sqrt{2}}$). So everything seems to agree with Gauss's note.
Despite what I wrote here, I do not have "real" understanding of why there is such a relation between spherical triangles and elliptic functions. Also, since Gauss himself wrote "Later note" in the beginning, it may came from a later period in his life, so it might not be a genuine result of him (I did not search for the history of this result).
Update
The connection of elliptic functions to spherical trigonometry is well-known, according to this Mathoverflow post. The first publications pointing on such a connection are those of A.M. Legendre and J.L. Lagrange from the years 1827-1828.
Gauss's notebooks on lemniscate elliptic functions are dated to the early period of 1797-1810, so it is possible that he precceeded them in developing this alternative geometrical point of view of elliptic functions. He seems to write this note in the context of justifying the addition theorem for elliptic functions, which is the same context that Legendre and Lagrange were operating in.