OK consistently there is indeed some Erdös-Rado type phenomenon in partition relations for linear orderings (so the bad order type as requested cannot be obtained by set forcing).
Assume there is a proper class of measurable cardinals.
Let $\kappa<\lambda$ be measurable cardinals, then ${\eta_\lambda \choose \eta_\kappa}\to {\eta_\lambda \choose \eta_\kappa}^{1,1}_\delta$ for any $\delta<\kappa$. We identify $(\eta_\lambda,<)$ with $(2^{<\lambda}, <_{lex})$, similarly for $2^{<\kappa}$. These are typical saturated linear orders.
Given $f: 2^{<\lambda}\times 2^{<\kappa}\to \delta$, for each $\sigma\in 2^{<\lambda}$, consider $f_\sigma: 2^{<\kappa}\to \delta$, by 1-dimensional Halpern-Läuchli theorem on $\kappa$ with $\delta$ many colors (denoted as $HL(1,\delta,\kappa)$, which is always true for measurable cardinals, see for example https://arxiv.org/abs/1608.00592), we know there exists a perfect strongly embedded subtree of $2^{<\kappa}$ such that $f_{\sigma}$ is monochromatic, say of color $\delta_\sigma$. As there are only $2^\kappa$ many strong subtrees of $2^{<\kappa}$, we can define a coloring $2^{<\lambda}\to 2^\kappa \times \delta$. By $HL(1,2^\kappa\times \delta,\lambda)$ we get a strong monochromatic subtree of $2^{<\lambda}$. Hence we have $T_0\subset 2^{<\lambda}$ strong subtree and $T_1\subset 2^{<\kappa}$ strong subtree such that there is some $\gamma\in \delta$ such that for each $\sigma\in T_0,\tau\in T_1$, $f(\sigma, \tau)=f_\sigma(\tau)=\delta_\sigma=\gamma$.
For any given order types $\theta, \psi$, we just need to find $\kappa, \lambda$ measurable as above large enough then it is always possible to embed $\theta, \psi$ into $\eta_\lambda, \eta_\kappa$ by saturation.