A non-closed leaf of a codimension-one foliation meets a closed transversal

Choose $x\in \bar{A}\setminus A$ and a product box centered at $x$. Infinitely many distinct plaques of $A$ meet smaller boxes about $x$; otherwise their finitely many transverse levels could not accumulate at the level of $x$ without including its plaque. Thus a short vertical segment $T$ meets $A$ twice. Join two such intersections by a compact embedded leafwise arc using F2 and removal of loops. Its intersections with $T$ are finite: they are closed in the compact arc and locally isolated by foliation boxes. Taking consecutive intersections along this arc gives a subarc whose interior misses $T$. Orient it from its higher endpoint to its lower endpoint.

Cover this compact arc by finitely many foliation boxes. Compose their plaque transports to obtain a thin foliated strip with central arc coordinate $u=0$ and positive transverse coordinate $u$; the transverse direction is consistent by F3. On this strip tilt the arc from $u=-\epsilon$ to $u=\epsilon$ with strictly positive derivative in $u$. Choose $\epsilon$ small enough that its final endpoint is still below its initial endpoint on $T$. Close it by the positive vertical segment between those endpoints. The central arc meets $T$ only at its endpoints, so a sufficiently thin strip and sufficiently small endpoint modifications make the closed curve embedded. It crosses $A$ where $u=0$, and every segment is positively transverse. Smooth the two corners inside product boxes. Convexity of the positive transverse tangent half-space preserves transversality, and a sufficiently small modification preserves embedding and the interior crossing of $A$.

The resulting curve is the required smooth embedded transverse circle meeting $A$. The construction uses finitely many boxes, one compact arc and finitely many shrinkings. It makes no arbitrary-neighborhood localization assertion.