Why is "be" in "this court rules he be put on probation" an infinitive?

I was watching an old movie, The Little Rascals, and one of the lines from a kid goes:

Your Honor, may I suggest… this court rules he be put on probation.

I am not sure if be is in infinitive form because this is an example of
subjunctive mood. If so, shouldn’t rules be the verb to be written as a infinitive because it is subordinate to suggest?

In addition, if I were to rewrite this sentence, I would write it as that it ran
something like this:

Your Honor, may I suggest… this court rule that he will be put on probation.

I’ve made two changes:

1. suggest (that) this court rules > suggest (that) this court rule
   (indicative > subjunctive/bare-infinitive)

2. (that) he be put > that he will be put
   (subjunctive/bare-infinitive > present-tense modal plus infinitive)

I wonder whether my rewrite would be considered grammatically correct.