J, 43 bytes

".e.-@-:@#((0<[:".{:\)*[:*/".\)"1 i.@!@#A.] 

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Takes digits as a string.

For each permutation i.@!@#A.], use half the digit length -@-:@# to take non-overlapping infixes -- which slices into 2 halves -- and multiply those halves([:*/".\)"1. Check if the input number ". is an element of that list e..

J, 43 bytes

".e.-@-:@#((0<[:".{:\)*[:*/".\)"1 i.@!@#A.] 

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Takes digits as a string.

For each permutation i.@!@#A.], use half the digit length -@-:@# to take non-overlapping infixes -- which slices into 2 halves -- and multiply those halves([:*/".\)"1. Then (to handle the trailing zero constraint) multiply that by (0<[:".{:\)*, which takes the last digits of each half, cats them, evaluates that as a two digit number, and checks if it's greater than 0.

Finally, check if the input number ". is an element of that list e..

J, 46 bytes

".e.-@-:@#([:((0<[:+/10&|)**/)".\)"1 i.@!@#A.] 

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Takes digits as a string.

For each permutation i.@!@#A.], use half the digit length -@-:@# to take non-overlapping infixes -- which slices into 2 halves -- and multiply those halves([:*/".\)"1. Check if the input number ". is an element of that list e..

J, 43 bytes

".e.-@-:@#((0<[:".{:\)*[:*/".\)"1 i.@!@#A.] 

Try it online!

Takes digits as a string.

For each permutation i.@!@#A.], use half the digit length -@-:@# to take non-overlapping infixes -- which slices into 2 halves -- and multiply those halves([:*/".\)"1. Check if the input number ". is an element of that list e..

J, 31 bytes

".e.-@-:@#([:*/".\)"1 i.@!@#A.] 

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Takes digits as a string.

For each permutation i.@!@#A.], use half the digit length -@-:@# to take non-overlapping infixes -- which slices into 2 halves -- and multiply those halves([:*/".\)"1. Check if the input number ". is an element of that list e..

J, 46 bytes

".e.-@-:@#([:((0<[:+/10&|)**/)".\)"1 i.@!@#A.] 

Try it online!

Takes digits as a string.

For each permutation i.@!@#A.], use half the digit length -@-:@# to take non-overlapping infixes -- which slices into 2 halves -- and multiply those halves([:*/".\)"1. Check if the input number ". is an element of that list e..