Retina, 8181 80 bytes
K`0. "$+"{+`(1(\.)|(1\.(0?1)*0?)1)(00|$) ${3}0${2}11 0\. 1. )+0`0?1(\.?)1 10${1}0 Try it online!Try it online! Works by repeatedly adding \$ 1 \$. Explanation:
K`0. Start off with \$ 0. \$.
"$+"{` )` Repeat the number of times given by the input.
+`(1(\.)|(1\.(0?1)*0?)1)(00|$) ${3}0${2}11 If the \$ \phi^0 \$ bit is set, use the identity \$ \phi^0 = \phi^{-1} + \phi^{-2} \$ to move bits away until there is enough room.
0\. 1. Add \$ 1 \$.
)+0`0?1(\.?)1 10${1}0 Minimise the number of bits by reversing the above identity.