Following Deane Yang's suggestion, it is not difficult to compute the Levi-Civita connection forms for the canonical orthonormal coframing of the orthogonal frame bundle. One then finds that such a $C$ depending on $C_1$ and $C_2$ with the desired properties does indeed exist and that there exist constants $a_0$, $a_1$, $a_2$, and $b$ depending only on the dimension $n$ such that $C$ can be taken to be
$$
C = a_0 + a_1\,C_1 + a_2\,{C_1}^2 + b\,C_2\,.\tag1
$$
When $n\le 2$, one can take $a_0=0$, but, when $n>2$, one must take all four of these constants to be positive in order to get an estimate that holds for all metrics in dimension $n$.

The point is that, as Deane says, one has a canonical coframing on $F(M)$ given by the tautological $1$-forms $\omega_i$ and the corresponding Levi-Civita $1$-forms $\omega_{ij} = -\omega_{ji}$ satisfying the first structure equation of Cartan (with the Einstein summation convention assumed here and below)
$$
\mathrm{d}\omega_i = -\omega_{ij}\wedge\omega_j\tag2
$$
and the second structure equation of Cartan
$$
\mathrm{d}\omega_{ij} = -\omega_{ik}\wedge\omega_{kj} + \tfrac12R_{ijkl}\,\omega_k\wedge\omega_l\,.\tag3
$$
The functions $R_{ijkl}=-R_{jikl}=-R_{ijlk}=-R_{iklj}-R_{iljk}$ on $F(M)$ represent the components of the Riemann curvature tensor of the original metric pulled back to $F(M)$ and satisfy
$$
\mathrm{d}R_{ijkl} =R_{pjkl}\,\omega_{pi}+R_{ipkl}\,\omega_{pj}+R_{ijpl}\,\omega_{pk}
+R_{ijkp}\,\omega_{pl}+R_{ijklm}\,\omega_m\tag4
$$
for unique functions $R_{ijklm}$, which represent the components of the covariant derivative of the Riemann curvature tensor pulled back to $F(M)$.

The canonical metric on the frame bundle is then given by
$$
g = \sum_{i=1}^n {\omega_i}^2 + \sum_{1\le i < j\le n}{\omega_{ij}}^2.\tag5
$$
If one uses the 'lexicographical' index ordering
$$
1 < 2<\cdots<n < 12 < 13\cdots <1n< 23 <\cdots < (n{-}1)n,
$$
for the $g$-orthonormal coframing
$$
\Omega = (\omega_1,\ldots,\omega_n,\omega_{12},\omega_{13},\ldots,\omega_{(n-1)1n}) = (\omega_\alpha)
$$
and writes $\mathrm{d}\omega_\alpha = -\theta_{\alpha\beta}\wedge\omega_\beta$, where $\theta_{\beta\alpha}=-\theta_{\alpha\beta}$, then one finds that for $n <\alpha<\beta$, the $1$-form $\theta_{\alpha\beta}$ is a constant linear combination of the $\omega_\gamma$
where $n<\gamma$; for $\alpha \le n < \beta$, the $1$-form $\theta_{\alpha\beta}$ is a constant linear combination of terms of the form $R_{\gamma\delta}\omega_\epsilon$, where $\epsilon \le n < \gamma,\delta$ (which, in particular, implies Deane Yang's assertion that the $\mathrm{O}(n)$-fibers of $F(M)\to M$ are totally geodesic in the metric $g$); while for $\alpha<\beta\le n$, the $1$-form $\theta_{\alpha\beta}$ is a linear combination of the $\omega_{\gamma}$ where $\gamma>n$ with coefficients that are *affine* linear combinations of the $R_{\gamma\delta}$ (where $\gamma,\delta>n$).

Substituting this information into the curvature formulae for the Levi-Civita connection $\theta$ for $g$, i.e.,
$$
\Theta_{\alpha\beta} = \mathrm{d}\theta_{\alpha\beta} + \theta_{\alpha\gamma}\wedge\theta_{\gamma\alpha}\,,\tag6
$$
and using the given formula for the exterior derivatives of the $R_{ijkl}$,
one finds that the coefficients of the $\Theta_{\alpha\beta}$ in the $\Omega$-coframing are linear combinations of terms that are either constants, constant multiples of $R_{ijkl}$, quadratic expressions in the $R_{ijkl}$ with constant coefficients, or constant multiples of the $R_{ijklm}$.

The existence of a constant $C$ bounding the sectional curvatures of $g$ that takes the form $(1)$ follows immediately from this.

**Example:** Take the case $n=2$. Then $F(M)\to M$ is an $\mathrm{O}(2)$-bundle and the structure equations on the $3$-manifold $F(M)$ become
$$
\mathrm{d}\omega_1 = -\omega_{12}\wedge\omega_2\qquad
\mathrm{d}\omega_2 = \omega_{12}\wedge\omega_1\tag{2'}
$$
and
$$
\mathrm{d}\omega_{12} = K\,\omega_1\wedge\omega_2\,,\tag{3'}
$$
where I have written $K$ for $R_{1212}$, as is traditional. ($K$ is simply the Gauss curvature.) The equation for the covariant derivative of the Riemann curvature tensor simply becomes, in this case.
$$
\mathrm{d}K = K_1\,\omega_1 + K_2\,\omega_2\tag{4'}
$$
Now, for simplicity and to avoid confusion, I am going to write $\omega_3$ for $\omega_{12}$, etc., so that
$$
\omega = \begin{pmatrix}\omega_1\\\omega_2\\\omega_3\end{pmatrix}
= (\omega_\alpha)
$$
becomes an orthonormal coframing for the metric $g = {\omega_1}^2+{\omega_2}^2+{\omega_3}^2$ on $F(M)$ whose curvature we want to compute. We do this by first finding the unique skew-symmetric $3$-by-$3$ matrix $\theta = (\theta_{\alpha\beta})$ that satisfies $\mathrm{d}\omega = -\theta\wedge\omega$. Given the equations $(2')$ and $(3')$, we find that
$$
\theta = (\theta_{\alpha\beta}) = \begin{pmatrix}
0 & (1{-}\tfrac12K)\omega_3 & -\tfrac12K\omega_2\\
-(1{-}\tfrac12K)\omega_3& 0 & \phantom{-}\tfrac12K\omega_1 \\
\tfrac12K\omega_2 & -\tfrac12K\omega_1& 0
\end{pmatrix},
$$
and the reader can verify that this has the general properties that I stated above for general $n$.

Now, we compute the curvature by computing the matrix
$$
\Theta = \mathrm{d}\theta + \theta \wedge\theta = (\Theta_{\alpha\beta}),
$$
and, using $(2')$, $(3')$, and $(4')$, we find that
$$
\begin{pmatrix} \Theta_{23} \\ \Theta_{31} \\ \Theta_{12}\end{pmatrix}
=
\begin{pmatrix}
\tfrac14K^2& 0 & -\tfrac12K_2 \\
0 & \tfrac14K^2 & \phantom{-}\tfrac12K_1\\
-\tfrac12K_2 & \phantom{-}\tfrac12K_1 & (K{-}\tfrac34K^2)\end{pmatrix}
\begin{pmatrix} \omega_2{\wedge}\omega_3 \\ \omega_3{\wedge}\omega_1 \\ \omega_1{\wedge}\omega_2\end{pmatrix}
$$
The Riemann curvature tensor of $g$ has now been shown to be
$$
\mathrm{Riem}(g) = \sum_{\alpha<\beta}\Theta_{\alpha\beta}\otimes \omega_\alpha{\wedge}\omega_\beta
= \Theta_{23}\otimes \omega_2{\wedge}\omega_3
+\Theta_{31}\otimes \omega_3{\wedge}\omega_1
+\Theta_{12}\otimes \omega_1{\wedge}\omega_2\,.
$$
It follows that the components of the Riemann curvature tensor of $g$ in this $g$-orthonormal coframing are linear combinations of $K$, $K^2$, $K_1$, and $K_2$. Thus, there is a bound on the sectional curvature of $g$ of the form claimed above in terms of $C_1$, an upper bound for $|K|$, and $C_2$, an upper bound for $\sqrt{{K_1}^2+{K_2}^2}$. (Note that, we do not need the constant $a_0$ in the case $n=2$. However, when $n>2$, constant terms do of course, show up, because the fibers of $F(M)\to M$, which are totally geodesic, are copies of $\mathrm{O}(n)$, which is not flat when $n>2$.)