= Solution
Integrating the exact Galerkin energy identity gives
$$
\frac12|u_m(t)|^2
+\nu\int_0^t\|u_m(\tau)\|^2\,d\tau
=\frac12|P_mu_0|^2
\leq\frac12|u_0|^2.
$$
Thus one may take
$$
\boxed{
K_0(T)=|u_0|,
\qquad
K_1(T)=\frac{|u_0|}{\sqrt{2\nu}}}
$$
for the first two requested bounds; these constants happen not to grow with $T$.
The Galerkin equation and contractivity of $P_m$ on $V'$ give
$$
\left\|\frac{du_m}{dt}\right\|_{V'}
\leq\nu\|Au_m\|_{V'}
+\|B(P_Nu_m,u_m)\|_{V'}.
$$
Since $\|Au_m\|_{V'}=\|u_m\|$, part a gives
$$
\left\|\frac{du_m}{dt}\right\|_{V'}
\leq
\left(\nu+c\lambda_N^{1/4}|u_m|\right)\|u_m\|
\leq
\left(\nu+c\lambda_N^{1/4}|u_0|\right)\|u_m\|.
$$
Consequently
$$
\boxed{
\left\|\frac{du_m}{dt}\right\|_{L^2(0,T;V')}
\leq
K_0'(T)
:=
\left(\nu+c\lambda_N^{1/4}|u_0|\right)
\frac{|u_0|}{\sqrt{2\nu}}}.
$$
All three constants are independent of $m$.
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