= Solution
First, all lifts of the real one-parameter projective group commute. Its scalar <commutator> $c(s,t)$ is again a continuous alternating bicharacter. For fixed nonzero $t$ and an integer $n>0$, a lift of $t$ is a phase multiple of the $n$th power of a lift of $t/n$; hence $c(t,t/n)=1$ and $c(t,mt/n)=1$ for every integer $m$. Rational multiples of $t$ are dense in $\mathbb R$, so continuity makes $c(t,s)=1$ for every $s$.
Choose a unitary $U$ representing $\rho(1)$ and the allowed <self-adjoint> logarithm $A\in U''$ with $U=e^{iA}$. Every lift of $\rho(t)$ commutes with $U$, hence with $A$ by the <double commutant theorem for star-algebras> condition. Therefore
$$
\widetilde\rho(t)=\rho(t)\,p(e^{-itA})
$$
is a continuous projective <homomorphism>, with $\widetilde\rho(1)=1$. It is periodic of period one and descends to $\mathbb R/\mathbb Z$. Part (b) lifts it to a continuous unitary <homomorphism> $L$ of that <circle>. Each $L(t)$ is a lift of $\widetilde\rho(t)$ and commutes with $A$, since a representative has the form of an original lift times $e^{-itA}$. Thus
$$
\boxed{\widehat U_t=L(t\bmod\mathbb Z)e^{itA}}
$$
is a strongly continuous unitary <homomorphism> with projective class $\rho(t)$. This gives the required lift of the whole real group, not just unrelated unitary representatives at each time.
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