JavaScript (Node.js), 100 bytes

s=>(i=1e4,g=(p,c)=>p[8]?i+=c&6?0:p<s:[0,7,1].map(n=>g(p+' xo'[n%5],~~c+n))|i)``.toString(2).slice(1)

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Input a string, output 13 char 0-1 string.

JavaScript (Node.js), 77 bytes

s=>(i=0,g=(p,c)=>p[8]?i+=c&6?0:p<s:[0,7,1].map(n=>g(p+' xo'[n%5],~~c+n))|i)``

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Input string as format 'x oxx o', output an integer in range \$\left[0,6105\right]\$.

JavaScript (Node.js), 100 bytes

s=>(i=1e4,g=(p,c)=>p[8]?i+=c&6?0:p<s:[0,7,1].map(n=>g(p+' xo'[n%5],~~c+n))|i)``.toString(2).slice(1)

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Input a string, output 13 char 0-1 string.

JavaScript (Node.js), 77 bytes

s=>(i=0,g=(p,c)=>p[8]?i+=c&6?0:p<s:[0,7,1].map(n=>g(p+' xo'[n%5],~~c+n))|i)``

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Input string as format 'x oxx o', output an integer in range \$\left[0,6105\right]\$.

We encoded all 6046 valid boards together with another 59 invalid ones into number not greater than 6105. The bit string output version also added an extra 1808 to each output so they actually encoded in range \$\left[1808, 7913\right]\$.

JavaScript (Node.js), 77 bytes

s=>(i=0,g=(p,c)=>p[8]?i+=c&6?0:p<s:[0,7,1].map(n=>g(p+' xo'[n%5],~~c+n))|i)``

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Input string as format 'x oxx o', output an integer in range \$\left[0,6105\right]\$.

JavaScript (Node.js), 100 bytes

s=>(i=1e4,g=(p,c)=>p[8]?i+=c&6?0:p<s:[0,7,1].map(n=>g(p+' xo'[n%5],~~c+n))|i)``.toString(2).slice(1)

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Input a string, output 13 char 0-1 string.

JavaScript (Node.js), 77 bytes

s=>(i=0,g=(p,c)=>p[8]?i+=c&6?0:p<s:[0,7,1].map(n=>g(p+' xo'[n%5],~~c+n))|i)``

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Input string as format 'x oxx o', output an integer in range \$\left[0,6105\right]\$.