Consider the moduli space $\cal{M}_{\hat g,d}$ of equivariant embeddings of surfaces (open, possibly non-oriented) into the quintic three-fold $X$ in $\mathbb{P}^4$ of a given degree $d \in H_2(X,L;\mathbb{Z})$ and genus $\hat g.$

(I must say that I'm not sure whether this has been rigorously defined previously.)

Here equivariance means the following: we can write $\Sigma = \hat\Sigma / \Omega$ where $\hat\Sigma$ is closed oriented of genus $\hat g$ and $\Omega$ is an orientation reversing involution; moreover we define
$\sigma:\mathbb{P}^4 \to \mathbb{P}^4$ given by $(x_1:x_2:x_3:x_4:x_5) \mapsto (\overline{x}_2:\overline{x}_1:\overline{x}_4:\overline{x}_3:\overline{x}_5)$ and call $L$ the pointwise fixed subset of $X$ under $\sigma.$ Equivariance means $f \circ \Omega = \sigma \circ f.$

I'd like to see (if this is true) in some generality that, for every map containing a crosscap (i.e. $\mathbb{RP}^2$) which develops a node on top of the lagrangian $L,$ this map admits another smoothing to a disk, and viceversa, thus giving a bijection between the two kinds of behavior.

A local model should be
$\mathbb{P}^1\ni(u:v) \mapsto (x:y:z) \in \mathbb{P}^2$
given by
$$ x= au^2, \quad y=av^2, \quad z=uv$$
which is mapped to the conic $xy-a^2z^2,$ which is real if $a^2 \in \mathbb{R}.$
This admits two different equivariant smoothings
$$ \begin{aligned}[3] &a \in \mathbb{R} \quad &(u:v)& \sim (\overline{v}:\overline{u}) \quad &\text{disk} \\
&a \in \mathrm{i}\mathbb{R} \quad &(u:v)& \sim (\overline{v}:-\overline{u}) \quad &\text{crosscap}\end{aligned}$$

Does this extend to higher genera (via Fenchel-Nielsen?)?

This is expected to hold only if $d$ contains an even ''piece'' $d_i,$
because only this can come from a trivial element in $H_1(L;\mathbb{Z})$ due to exactness in
$$ H_2(X;\mathbb{Z}) \to H_2(X,L;\mathbb{Z}) \to H_1(L;\mathbb{Z})$$