The Figiel-Johnson Tsirelson space is an example of an asymptotic $\ell_1$ Banach space not containing $\ell_1$. The notion of asymptotic $\ell_1$ is with respect to some basis, but a coordinate free notion exists (sometimes referred to Asymptotic $\ell_1$ in the literature, although I am not sure whether this terminology is widespread). This can be defined via weakly null trees of height $n$, or in terms of a two player game. For fixed $n$ and $K>0$, Player I chooses a weak neighborhood $U_1$ of $0$, Player II chooses $x_1\in S_X\cap U_1$, etc., until $U_1, \ldots, U_n$ and $x_1, \ldots, x_n\in S_X$ are chosen. Player I wins if $(x_i)_{i=1}^\infty$ is $K$-equivalent to the $\ell_1^n$ basis. Then we say $X$ is Asymptotic $\ell_1$ if there exists $K>1$ such that for every $n\in\mathbb{N}$, Player I has a winning strategy in the game with this $n$ and $K$.
What I am looking for is: For every limit ordinal $\xi$, if $c_{00}(\xi)$ denotes the span of the Hamel basis $(e_\gamma: \gamma<\xi)$, can we define a norm $|\cdot|$ on $c_{00}(\xi)$ to have the following properties?
$(1)$ $(e_\gamma:\gamma<\xi)$ is normalized and $1$-unconditional.
$(2)$ The completion $X$ of $c_{00}(\xi)$ with respect to $|\cdot|$ is Asymptotic $\ell_1$.
$(3)$ There does not exist a countably infinite subset $G$ of $\xi$ such that $(e_\gamma)_{\gamma\in G}$ is equivalent to the $\ell_1$ basis.
Note that we do not require that $X$ contains no copy of $\ell_1$, only that sequences in the basis do not give you copies of $\ell_1$.