That's a good question and is not exactly easy to answer. The Einstein field equations of General Relativity are highly non-linear which makes it difficult to talk about superpositions of solutions to the equations and to compare local physics (at the level of black holes) to cosmological effects (e.g. the expansion of the universe) and transfer results and insights between them.
Let me explain. When we talk about cosmology in general and the (accelerated) expansion of the universe in particular, we zoom out to very large scales and consider a homogenous model of the universe (a so-called FLRW spacetime[0]). Here, homogeneity means that mass density, hubble rate and so on are the same across the universe and only depend on time.
We then deduce that at this scale and under these assumptions we need to incorporate an additional parameter Λ, called cosmological constant aka dark energy, into our field equations / cosmological solution in order to match observations.[1]
Homogeneity was a simplifying assumption, though! Our universe is clearly not homogeneous! Next to your head, there is air, and inside your head evidently not :), and so the mass density is clearly not constant across space!
The same thing holds for black holes: Outside a black hole there's vacuum, inside a black hole there is… Well, we don't really know but the mass from which it formed has gotta be somewhere, right? So the mass density in black hole spacetimes is presumably not constant, either.
Interestingly, we know that spacetime near other celestial bodies (galaxies, stars, planets, moons, …) can be approximately described by one of the black hole spacetimes, too. (This is because outer region of black hole spacetimes describes not just black holes but any spherically or axially symmetric static/stationary spacetime.) So it's turtl—… uhh outer black hole spacetimes all the way down!
Anyway, in those cases of stars/planets/moons we have some massive object in the center of the spacetime region and vacuum outside – once more, the mass density is clearly non-zero!
So how do we square this with the homogeneity assumption of our cosmolical model (ΛCDM)? We can't but that's perfectly fine from a logical point of view. The reason is that we cannot simply take all those local spacetimes (of all stars, planets and black holes) and simply add them up to obtain the spacetime of the entire universe, because the field equations are not linear. Adding up two solutions does in general not yield a third solution! Conversely, we cannot simply zoom in on the FLRW spacetime and then compare a local, perfectly homogeneous "snippet" of FLRW with a black hole spacetime – this comparison does not make sense a priori.
Unfortunately, the reality is we simply don't know how to zoom in / zoom out between different spacetimes at different scales. So while we have a cosmological constant at large scales (when assuming homogeneity), there is no guarantee such a constant makes sense at smaller scales, i.e. that there is expansion at smaller scales.
This is because there are two possible ways to interpret the cosmological constant: One way is to interpret it as a fixed parameter in the field equations (i.e. it influences every solution), another way is to interpret it as a term in the specific large-scale spacetime solution of our universe (i.e. ΛCDM) and shove it into that solution's energy-momentum tensor.
Now, I think most physicists adhere to the first interpretation and assume that the constant is the same across all scales and possibly even homogeneous. Thus it should impact all spacetime solutions in the same way and at all scales, including black hole solutions.
For this reason people have introduced modified black hole solutions that, like our cosmological model, take into account a positive cosmological constant (= "de Sitter"), for instance
https://en.wikipedia.org/wiki/De_Sitter%E2%80%93Schwarzschil...
As you suspected, in these solutions you generally have both effects that counteract each other: The gravitational pull of the black hole (leading to an event horizon) and the expansion of the universe ("anti-gravitational pull") due to the cosmological constant, leading to a cosmological horizon in the region far away from the black hole.
Nevertheless, whether you believe in/consider a cosmological constant at small scales is a bit up to you. Most people I know seem assume that space at the level of atoms, planets, solar systems or even galaxies is not expanding and only very far away from gravitational systems you'll end up with expansion. This is supported by the fact that the cosmological constant is so tiny and, thus, in the aforementioned De Sitter black hole solutions, the cosmological horizon is far, far away from the center – so far indeed (111 (M/Msolar)1/3 parsecs[2]) that we know this outside region of the spacetime can no longer be valid/applicable.
To see the latter, remember that black hole solutions assume a perfect vacuum away from the central body (black hole/star/planet/…) but in reality no massive body is alone in the universe. This means that, before you reach a distance of 111 (M/Msolar)1/3 parsecs from a given body, you'll long have encountered another couple massive bodies which will modify your spacetime and cause additional gravitational pull. (Once again, how exactly they modify spacetime we don't know (short of maybe some numerical approximations), since we can't easily superpose spacetimes!)
Long story short: Only in very few situations it's worth considering a cosmological constant at small scales. Its effects are easily cancelled out / hidden by local gravity.
[0]: https://en.wikipedia.org/wiki/Friedmann%E2%80%93Lema%C3%AEtr...
[1]: https://en.wikipedia.org/wiki/Lambda-CDM_model
[2]: https://ui.adsabs.harvard.edu/abs/2020AAS...23537904F/abstra...