How to benchmark #2-SAT counting algorithms? Are there any libraries of #2-SAT instances that are hard to solve with state-of-the-art exact solvers (sharpSAT, cachet, ...)?

Alternatively: are there practical ways to generate hard #2-SAT instances?

I have only been able to find benchmark instances for #k-SAT for $k\ge3$.
I am not aware of any collections of 2CNF benchmark instances.

However, one practical way of constructing #2-SAT instances that are provably hard for state-of-the-art model counters is as follows: generate a $d$-regular expander for some constant $d$ (so in practice a random $d$-regular graph), then, as holf suggested, construct the 2CNF-formula that has for every edge $uv$ the clause $x_u\lor x_v$. By this paper

https://arxiv.org/abs/1411.1995

of Bova et al, the resulting formulas do not have small circuits of the form that is called DNNF in knowledge compilation, as subfield of artificial intelligence. Then by connections between DPLL-based model counting and DNNF that can be found in 

Huang, Jinbo, and Adnan Darwiche. "DPLL with a trace: From SAT to knowledge compilation." IJCAI. Vol. 5. 2005.

it follows that all DPLL-based model counters will take exponential time to solve the above instances. Since all exact model counters that I am aware of (except for this knowledge compiler that can also used for counting http://www.cril.univ-artois.fr/KC/eadt.html ) are based on exhaustive DPLL, this should answer your alternative question. 

Note that the results of Bova et al. are rather asymptotic (there are some small hidden constants in the analysis) but from my experience with similar problems, actual implemented model counters should fail quite early on those instances. 

Note that these instances might not be the ones you are looking for: instead of challenging state-of-the-art model counters, they just outright break them provably. So to work around this, you would have to completely change the approach from DPLL to something else similarly to the compiler I linked above (which I don't think will work on these instances, either).