SFT compactness is, as far as I know, in this situation not directly applicable, since the genera of the curves $u_n$ may be unbounded (maybe I am missing something which ensures genus bounds for them in this situation, or one can use the additional assumptions to make SFT compactness applicable).
In any case, there is in dimension 4 a compactness result for $J$-holomorphic currents by Taubes, which is often used for ECH, and could presumably be used here.

Assuming for the moment genus bounds on the $u_n$, SFT compactness shows that a subsequence of $u_n$ converges to a holomorphic building $v$, that is "a curve" with potentially several levels $v=(v_1=:v_+, ...,v_k=:v_-)$.

If there is no splitting ($k=1,v_+=v_-$), then SFT compactness ensures that a subsequence of curves converges (up to shifts in the $\mathbb{R}$-direction) to a curve with the same asymptotes, that is to some $[v]\in \mathcal{M}^J(\alpha_+,\alpha_-)/\mathbb{R}$.

If there is a splitting ($k>1$), note first that the assumptions relating to "parametrization by $C_\pm$" ensure that the parts of $u_n$, whose image is parametrized by $C_\pm$, are both contained in only one level (that is, in this region there is no further splitting possible). If we use then in addition the assumption on $C_0$, we can also conclude that the lowest level $v_-$ has the same asymptotes as $u_-$ and the top level $v_+$ has the same asymptotes as $u_+$ and
in the middle we have a piece (consisting
potentially of none, one or several levels), whose top asymptotes
coincide(!) with the bottom asymptotes (since these coincide with the asymptotes of $u_{\mp}$ (or also of $v_\mp$) at $\pm\infty$ ).

Here we used throughout, that $\delta$ can be chosen so small, such that the $\delta$-neighborhood $N_\delta$ of the asymptotic orbits deformation retracts to this collection of the asymptotic orbits.
In other words the asymptotes (up to time shifts) are determined by these $C^0$-data.

It follows that the $d\alpha$ energy of the whole middle piece is 0
(the $d\alpha$-energy can be computed as difference of the periods of the asymptotic orbits, and depends only on the relative homology class).
Any proper nonconstant curve of finite Hofer energy with $d\alpha$ energy zero is necessarily a branched cover over a collection of cylinders. Thus the whole middle piece is a branched cover over a collection of cylinders (potentially in different middle levels).

It remains to identify the top and the bottom component as $u_\pm$
(we know already that the images have the same asymptotes).
Since the asymptotes are given and the $d\alpha$-energy is computed from the asymptotes, it follows that the sum of the $d\alpha$-energies of the *simple curves* underlying $v_\pm$ coincides with the sum of $d\alpha$ energies of $u_\pm$.
Since moreover the $d\alpha$ energy converges (and the $d\alpha$ energy is additive over the levels), the $d\alpha$ energy of $v$
is equal to the $d\alpha$-energy of the $u_n$'s, which is in turn the same
as the sum of the $d\alpha$-energies of $u_\pm$.

Therefore all of the components of $v_\pm$, which contribute $d\alpha$-energy, are simple, i.e. they are not nontrivial branched covers of other curves. Thus at most trivial cylinders in the image of $v_\pm$ are (branched) covered.
On the other hand, condition 1d ensures now exactly that any (possibly branched) cover over the trivial cylinders with the given asymptotes $(a_i,b_i ...)$ are necessarily unbranched, and thus these components are determined as maps by the asymptotes; hence the components in $v_\pm$ which are (apriori branched) covers of trivial cylinders coincide with the unbranched covers occuring in $u_\pm$.

The components of $v_\pm$ contributing nonzero $d\alpha$-energy are therefore parametrized by surfaces of fixed topological type (the same as for $u_\pm$), and we can (as you mentioned) choose $\delta$ small enough, to ensure that also these components coincide (as parametrized curves up to translation)
with the corresponding components of $u_\pm$,
as the $[u_\pm]$ are isolated in their moduli space.

**Edit**: More on why the ends of $u_+$ coincide with the ends of $v_+$.
First I should be more precise above (in paragraph 4) to say, that at that point one only knows, that the ends of $u_+$
coincide (with multiplicity) with the the ends of the *simple* curve underlying $v_+$.

This more precise statement should hold for the following reasons:
First, all negative ends of $u_+$ can be distinguished on a $\delta$-level in the target (as $u_+$ is simple and by "unique continuation as almost complex submanifold" or perhaps more directly studying the asymptotes in coordinates). Then there is for each end $e_u$ of $u_+$ negatively asymptotic to some Reeb orbit (almost parametrized by some circle $c$ in $e_u$), a circle $c'$ in the curve $v_+$ (parametrized via $\psi_+$ by $c \subset e_u$), which lies in the same component of the $\delta$-neighborhood $N_\delta$ and is homotopic in $N_\delta$ to $c$. Then the "parametrized component" of $c'$ in $N_\delta \cap v_+$ does not leave $N_\delta$ again by the assumption on $C_0$. Thus $c'$ determines an end $e_v$ of the simple curve underlying $v_+$, which wraps around the Reeb orbit like $c'$ and thus with the same multiplicity as $e_u$. This procedure works for all ends of $u_+$ and all ends of the simple curve underlying $v_+$ are among these.

Then one proceeds with the discussion as above, using only what we know about the asymptotics of the simple curves underlying $v_\pm$.