As mentioned by David White in the comment, I've recently proved that left induced model structure exists for any "tractable" class of cofibrations on a locally presentable category.

Tractable means that the class of cofibration is generated by a set of cofibration with cofibrant domain. It seems to me that the $\partial [0,1]^n$ can be obtained by gluing of their faces and so are cofibrant, so I think this applies to your category.

However I should also say that my construction isn't much more explicit than the one using Vopenka's principle, so I'm not sure it is very helphful for your question: it does not tell you anything about the model structure structure itself, like a nice cylinder functor would, except maybe a bound on the size of the generating trivial cofibrations.

A bit more precisely, you consider $W$ the "smallest Cisinski localizer", i.e. the smallest class of maps containing "trivial fibrations" closed under 2-out-of-3 and such that $W$ intersected with your class of cofibration is clsoed under pushout and transfinite composition, and you take for J (the generating trivial cofibrations) all cofibrations that are in $W$ and between $\kappa$-presentable object for some large enough cardinal $\kappa$ (I think in your case $\kappa = 2^\omega$ might work, though it might be $(2^\omega)^+$ depending on what is the presentability rank of the $[0,1]^n$ in your category, which I'm not quite sure about).

Then you have a model structure with $W$ as the weak equivalences, your cofibrations and $J$ as the generating trivial cofibration.

As mentioned by David White in the comment, I've recently proved that left induced model structure exists (without any kind of large cardinal axiom) for any "tractable" class of cofibrations on a locally presentable category.

Tractable means that the class of cofibration is generated by a set of cofibration with cofibrant domain. It seems to me that the $\partial [0,1]^n$ can be obtained by gluing of their faces and so are cofibrant, so I think this applies to your category.

However I should also say that my construction isn't much more explicit than the one using Vopenka's principle, so I'm not sure it is very helphful for your question: it does not tell you anything about the model structure structure itself, like a nice cylinder functor would, except maybe a bound on the size of the generating trivial cofibrations.

A bit more precisely, you consider $W$ the "smallest Cisinski localizer", i.e. the smallest class of maps containing "trivial fibrations" closed under 2-out-of-3 and such that $W$ intersected with your class of cofibration is clsoed under pushout and transfinite composition, and you take for J (the generating trivial cofibrations) all cofibrations that are in $W$ and between $\kappa$-presentable object for some large enough cardinal $\kappa$ (I think in your case $\kappa = 2^\omega$ might work, though it might be $(2^\omega)^+$ depending on what is the presentability rank of the $[0,1]^n$ in your category, which I'm not quite sure about).

Then you have a model structure with $W$ as the weak equivalences, your cofibrations and $J$ as the generating trivial cofibration.

As mentioned by David White in the comment, I've recently proved that left induced model structure exists for any "tractable" class of cofibrations on a locally presentable category.

Tractable means that the class of cofibration is generated by a set of cofibration with cofibrant domain. It seems to me that the $\partial [0,1]^n$ can be obtained by gluing of their faces and so are cofibrant, so I think this applies to your category.

However I should also say that my construction isn't much more explicit than the one using Vopenka's principle, so I'm not sure it is very helphful for your question: it does not tell you anything about the model structure structure itself, like a nice cylinder functor would, except maybe a bound on the size of the generating trivial cofibrations.

A bit more precisely, you consider $W$ the "smallest Cisinski localizer", i.e. the smallest class of maps containing "trivial fibrations" closed under 2-out-of-3 and such that $W$ intersected with your class of cofibration is clsoed under pushout and transfinite composition, and you take for J (the generating trivial cofibrations) all cofibrations that are in $W$ and between $\kappa$-presentable object for some large enough cardinal $\kappa$ (I think in your case $\kappa = 2^\omega$ might work, though it might be $2^(2^\omega)$ depends on what is the presentability rank of the $[0,1]^n$ in your category, which I'm not quite sure about).

Then you have a model structure with $W$ as the weak equivalences, your cofibrations and $J$ as the generating trivial cofibration.

As mentioned by David White in the comment, I've recently proved that left induced model structure exists for any "tractable" class of cofibrations on a locally presentable category.

Tractable means that the class of cofibration is generated by a set of cofibration with cofibrant domain. It seems to me that the $\partial [0,1]^n$ can be obtained by gluing of their faces and so are cofibrant, so I think this applies to your category.

However I should also say that my construction isn't much more explicit than the one using Vopenka's principle, so I'm not sure it is very helphful for your question: it does not tell you anything about the model structure structure itself, like a nice cylinder functor would, except maybe a bound on the size of the generating trivial cofibrations.

A bit more precisely, you consider $W$ the "smallest Cisinski localizer", i.e. the smallest class of maps containing "trivial fibrations" closed under 2-out-of-3 and such that $W$ intersected with your class of cofibration is clsoed under pushout and transfinite composition, and you take for J (the generating trivial cofibrations) all cofibrations that are in $W$ and between $\kappa$-presentable object for some large enough cardinal $\kappa$ (I think in your case $\kappa = 2^\omega$ might work, though it might be $(2^\omega)^+$ depending on what is the presentability rank of the $[0,1]^n$ in your category, which I'm not quite sure about).

Then you have a model structure with $W$ as the weak equivalences, your cofibrations and $J$ as the generating trivial cofibration.