Question: Can we have a model of $ZF-\text {Regularity}$ where there exist an ordinal $\kappa$ such that $H_{\kappa}$ exists and $H_{\kappa}$ is not equinumerous to any well founded set?

The motivation for this question comes in connection with defining Cardinality under some situations beyond Regularity and Choice. Especially in connection with the following anti-foundation axiom:

$\text{Anti-Foundation axiom: }$ Every set is subnumerouse to some iterative power of some $H_\kappa$ set, where $\kappa$ is an ordinal.

where $H_\kappa = \{x| x \text{ is hereditarily subumerous to } \kappa\}$

Where $x \text{ is hereditarily subnumerous to } \kappa $ is defined as: $$ \forall y \in TC(\{x\} ) \exists f (f: y \rightarrowtail \kappa)$$

Where $TC$ stands for the "transitive closure function" defined in the usual manner.

Iterative powers $P^i(x)$ are defined recursively as:

$P^0(x) = x$

$P^j(x) = \bigcup (\{P (P^i(x))| i < j\})$

Using this as an anti-foundation axiom would enable us to define a notion for Cardinality that covers more sets than does Scott's definition of Cardinality.

$\text{Define:} $ Card(x) is the set of all subsets of the first supernumerous to $x$ iterative power of the nearest $H_{\kappa}$ set to $x$, that are equinumerous to $x$.

The distance of $x$ from $H_{\kappa}$ is the minimal ordinal $i$ such that $P^i(H_{\kappa})$ supernumerous to $x$.

Of all $H_{\kappa}$ sets that lie at the least distance from $x$, the one with the least $\kappa$ value is the "Nearest $H_{\kappa}$ set to $x$".

The idea is that there is no combinatorial restriction on what constitutes the cardinality of an $H_{\kappa}$ set, so this definition can work without having a practical kind of restriction over the non-well ordered, non-well founded realm. While Scott's definition can only define cardinality for sets as long as those are equinumerous to some well founded set. Which is in some sense restrictive outside of choice.