The first question appears to have an affirmative answer in Corollary 1 in Exotic Twisted Equivariant Cohomology of Loop Spaces, Twisted Bismut–Chern Character and T-Duality by Fei Han and Varghese Mathai. However this seems to conflict with the answer by Tom Goodwillie, and I am not yet able to exactly see what is causing this discrepancy. My guess is that actually two different versions of $LM$ as a topological space/infinite dimensional manifold occur in the two articles so that both statements are true relative to the version of $LM$ they assume. But this is just a quick guess.

As the example by Tom Goodwillie shows, the restriction to fixed points does not generally induce an isomorphism between localized equivariant cohomologies. Yet, as the question shows, there's good reason for willing this isomorphism to exist despite the evidence it does not. The question has been solved by Jones and Petrack in The fixed point theorem in equivariant cohomology. The following extract from their Introduction beautifully explains what they do:

"Now let $X$ be a smooth, connected, finite dimensional manifold and let $LX$ be the space of all smooth loops in $X$ with its $C^\infty$ topology. The space $LX$ is an infinite dimensional manifold modelled on a Fr'echet space. The circle acts smoothly on LX and the fixed point set of this action is precisley the manifold X, considered as the space of constant loops. According to Goodwillie [10], $u^{-1}H^\bullet_T(LX)$ depends only on $\pi_1(X)$ so that the fixed point theorem, as it stands, cannot be true for $LX$. This has been an obstacle to progress in the study of differential forms and integration on $LX$. Our solution to this difficulty is to construct a new form of equivariant cohomology, denoted $h^\bullet_T(Y)$, which will be used in place of $u^{-1} H^\bullet_T(Y)$. The groups $h^\bullet_T(Y)$ are defined by a simple and natural modification of one definition of periodic equivariant cohomology. If $Y$ is finite dimensional then $h^\bullet_T(Y)=u^{-1} H^\bullet_T(Y)$ but in a large class of infinite dimensional examples, including the case $Y = LX$ , the inclusion $i: F \to Y$ of the fixed point set induces an isomorphism $h^\bullet_T(Y) \to h^\bullet_T(F)$. This cohomology theory is constructed in §1 and the fixed point theorem is stated precisely in §2. Some alternative theories which also satisfy the fixed point theorem are discussed in §3. In §4 we show how to construct an inverse, at the level of differential forms, for the restriction homomorphism $i^\ast$ and following [3, 6, 4, and 7] explain how this leads to integration formulas. The most important ingredient is the construction of an equivariant form $\tau$, essentially the form $\exp( -( \omega + E))$ of [2, 6], which defines a class in $h^\bullet_T(Y)$ and has the key property that $i^\ast(\tau) = 1$. This form $\tau$ is essentially the equivariant Thom/Euler class of the normal bundle to the fixed point set. In §5 we show how the form $\tau$ on $LX$, is related to the $\hat{A}$ polynomial of $X$, compare [2]."