Since you've referenced the paper by Delaney, Rowell and Wang (where the authors talk about braided fusion categories), I'll add a comment (in response to your question about taking more interesting quotients) on how 'higher order' Hecke algebras naturally appear in unitary braided fusion categories, and how tracing then yields framed link invariants.

Definition: Let's define a (generalised) Iwahori-Hecke algebra $H_{n}(Q,k)$ to be the quotient of braid algebra $\mathbb{C}[B_{n}]$ by an ideal generated by $\{\Pi_{i=1}^{k}(\sigma_j - r_{i})\}_{j=1}^{n-1}$ where $r_{i}\in\mathbb{C}^{\times}$ are fixed, and $k\geq2$. The case $k=2$ corresponds to the usual Iwahori-Hecke algebra.

Let $\mathcal{C}$ be a unitary braided fusion category, and $Irr(\mathcal{C})$ denote a set of representatives of isomorphism classes of simple objects in $\mathcal{C}$. Take some self-dual $q\in Irr(\mathcal{C})$ such that $q\otimes q \cong \bigoplus_{i=1}^{k} x_{i}$ where $k\geq 2$ and the simple subobjects $\{x_{i}\}_{i}$ are all distinct elements of $Irr(\mathcal{C})$. It turns out that the action of the $n$-strand braid group $B_{n}$ on $End(q^{\otimes n})$ defines a unitary representation $\rho$ of the generalised Iwahori-Hecke algebra $H_{n}(Q, k)$, where $\{r_{i}\}_{i=1}^{k}$ are the eigenvalues of the unitary braiding matrix $R^{qq}\in U(End(q\otimes q))$. If you now take an arbitrary $n$-braid $f_{n} \in End(q^{\otimes n})$ and perform the trace, you will have a link diagram that evaluates to a scalar $tr(f_{n})\in\mathbb{C}$ which is invariant under braid isotopy.

In the above, the framed link invariant corresponding to the case $k=2$ is the Kauffman bracket. For $k=3$, it's the Dubrovnik or Kauffman polynomial -- these can be seen as coming from the 'cubic' Hecke algebra $H_{n}(Q,3)$. In both cases, the representation $\rho$ will also factor through the Temperley-Lieb algebra. Check out the paper [1] (specifically Theorem 3.1) by Morrison, Peters and Snyder. However, I'm not sure if there are results on classifying invariants this way for $k>3$.$^{\dagger}$

[1] https://arxiv.org/abs/1003.0022

$\dagger $ A partial result for $k=4$ is claimed (yet to appear?) in [1] that corresponds to Kuperberg's G2 invariant.

Since you've referenced the paper by Delaney, Rowell and Wang (where the authors talk about braided fusion categories), I'll add a comment (in response to your question about taking more interesting quotients) on how 'higher order' Hecke algebras naturally appear in unitary braided fusion categories, and how tracing then yields framed link invariants.

Definition: Let's define a (generalised) Iwahori-Hecke algebra $H_{n}(Q,k)$ to be the quotient of braid algebra $\mathbb{C}[B_{n}]$ by an ideal generated by $\{\Pi_{i=1}^{k}(\sigma_j - r_{i})\}_{j=1}^{n-1}$ where $r_{i}\in\mathbb{C}^{\times}$ are fixed, and $k\geq2$. The case $k=2$ corresponds to the usual Iwahori-Hecke algebra.

Let $\mathcal{C}$ be a unitary braided fusion category, and $Irr(\mathcal{C})$ denote a set of representatives of isomorphism classes of simple objects in $\mathcal{C}$. Take some self-dual $q\in Irr(\mathcal{C})$ such that $q\otimes q \cong \bigoplus_{i=1}^{k} x_{i}$ where $k\geq 2$ and the simple subobjects $\{x_{i}\}_{i}$ are all distinct elements of $Irr(\mathcal{C})$. It turns out that the action of the $n$-strand braid group $B_{n}$ on $End(q^{\otimes n})$ defines a unitary representation $\rho$ of the generalised Iwahori-Hecke algebra $H_{n}(Q, k)$, where $\{r_{i}\}_{i=1}^{k}$ are the eigenvalues of the unitary braiding matrix $R^{qq}\in End(q\otimes q)$. If you now take an arbitrary $n$-braid $f_{n} \in End(q^{\otimes n})$ and perform the (quantum) trace $\widetilde{Tr}$, you will have a link diagram that evaluates to a scalar $\widetilde{Tr}(f_{n})\in\mathbb{C}$ which is invariant under braid isotopy. (N.B. the quantum trace can be understood as a weighted matrix trace, see attached notes).

In the above, the framed link invariant corresponding to the case $k=2$ is the Kauffman bracket. For $k=3$, it's the Dubrovnik or Kauffman polynomial -- these can be seen as coming from the 'cubic' Hecke algebra $H_{n}(Q,3)$. In both cases, the representation $\rho$ will also factor through the Temperley-Lieb algebra. Check out the paper [1] (specifically Theorem 3.1) by Morrison, Peters and Snyder. However, I'm not sure if there are results on classifying invariants this way for $k>3$.$^{\dagger}$

Attachment: In response to OP's request, rough notes explaining how representations of the Iwahori-Hecke algebra arise in unitary braided fusion categories can be found at [2].

[1] https://arxiv.org/abs/1003.0022

[2] https://sites.google.com/view/sachinvalera/rough-notes

$\dagger $ A partial result for $k=4$ is claimed (yet to appear?) in [1] that corresponds to Kuperberg's G2 invariant.