If you fix $N$ and group elements $g_1$, ..., $g_N \in G$, then your question becomes closely related to tilings of groups. Specifically, in Chapter 2, section 2 of "[Entropy and Isomorphism Theorems for Actions of Amenable Groups][1]," Ornstein and Weiss prove:

Let $G$ be a countable group acting freely and measure preservingly on a standard probability space $(X, \mu)$. Fix a finite set $T \subseteq G$. If for every $\epsilon > 0$ there is a measurable set $U \subseteq X$ such that the $T$-translates of $U$ are disjoint and $\mu(T \cdot U) > 1 - \epsilon$, then $T$ tiles $G$ in the sense that there is a set of centers $C \subseteq G$ such that the sets $Tc$ ($c \in C$) partition $G$.

They also prove that if $G$ is amenable then the converse holds. So if $G$ is amenable and $T$ is a tile for $G$, then for every free probability measure preserving action of $G$ and every $\epsilon > 0$ there is a $(\epsilon, |T|)$-fundamentalish domain. Thus a natural question is: which amenable groups admit arbitrarily large finite tiles?

Weiss called a group $G$ MT (mono-tileable) if for every finite set $F \subseteq G$ there is a finite tile $T \subseteq G$ containing $F$. In "[Monotileable amenable groups][2]," Weiss proved that all solvable groups and all residually finite groups are MT. In "[Elementary Amenable Groups][3]," Chou proved that all elementary amenable groups and all free products of non-trivial groups are MT. So in particular, your question has a positive answer whenever the group $G$ is elementary amenable. A stronger tiling condition, called ccc, is studied in chapter 4 of "[Groups Colorings and Bernoulli Subflows][4]" (this paper is in preparation). Weaker properties of poly-MT and poly-ccc are studied in "[Burnside's Problem, spanning trees, and tilings][5]." To the best of my knowledge these are the only papers which study tilings of countable groups.


  [1]: https://doi.org/10.1007/BF02790325 "J. Anal. Math. 48, 1–141 (1987). zbMATH review at https://zbmath.org/0637.28015"
  [2]: https://books.google.com/books?id=NDJ0rRuSMScC&pg=PA257&lpg=PA257&dq=monotileable+amenable+groups&source=bl&ots=DpqekXyi_H&sig=ShnrG6LH5nqIqT1QKGWVcxJFXlM&hl=en&ei=aovBTs3_D8W02wWAs-GxBQ&sa=X&oi=book_result&ct=result#v=onepage&q=monotileable%20amenable%20groups&f=false "Turaev, V. (ed.) et al., Topology, ergodic theory, real algebraic geometry. Rokhlin’s memorial. Providence, RI: American Mathematical Society (AMS). Transl., Ser. 2, Am. Math. Soc. 202(50), 257–262 (2001). zbMATH review at https://zbmath.org/0982.22004"
  [3]: https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-24/issue-3/Elementary-amenable-groups/10.1215/ijm/1256047608.full "Ill. J. Math. 24, 396–407 (1980). zbMATH review at https://zbmath.org/0439.20017"
  [4]: https://web.archive.org/web/20120204105310/http://www-personal.umich.edu/~bseward/Files/Group%20Colorings%20and%20Bernoulli%20Subflows.pdf "Gao, Su; Jackson, Steve; Seward, Brandon. Group colorings and Bernoulli subflows. Mem. Am. Math. Soc. 1141, vi, 246 p. (2016). doi:10.1090/memo/1141. arXiv:1201.0513. zbMATH review at https://zbmath.org/1375.37036"
  [5]: https://web.archive.org/web/20120204105316/http://www-personal.umich.edu/~bseward/Files/Burnside's%20Problem,%20Spanning%20Trees,%20and%20Tilings.pdf "Seward, Brandon. Burnside’s problem, spanning trees and tilings. Geom. Topol. 18, No. 1, 179-210 (2014). doi:10.2140/gt.2014.18.179. arXiv:1104.1231. zbMATH review at https://zbmath.org/1338.20041"