Just a suggestion. Spelled "theatre" for UK/Canada, IIRC.

www.vam.ac.uk/content/articles/0-9/17th-century-theatre/

From the way it is being used in your text, I think this would be sufficient; and as Helen has shown us, no I don't think we have a speific term for it in this period in England: French and Italian theatre was far more technically sophisticated in these early times. Which is one reason so much of our modern terminology comes, directly or indirectly, from those languages.

As an alternative, you might use 'mechanized theatre', to convey the idea that it uses sophisticated (for the period!) machinery.

For some perhaps rather later examples, do watch Ingmar Bergman's "Die Zauberflöte" using the traditional (original!) stage workings from Drottningholm Palace.

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Note added at 55 mins (2019-01-06 22:12:10 GMT)
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Here's an article that might make for interesting reading — it seems that 'castelet' may have been a term sometimes used for a marionnette theatre; I'm thinking rather like an ancestor of the traditional 'Punch & Judy' 'tent' theatres we have in Britain?

BaroquiadeS -- Atys en Folie - d'après J.B. Lully

www.baroquiades.com/.../1/atys-en-folie-lully-desrousseaux-...

La machinerie du castelet, véritable théâtre dans le théâtre, permet de passer d'un décor à un autre en un éclair sous nos yeux véritablement ...

The sentence will work much better if you use a single term here rather than a phrase, and I think it's perfectly feasible. I have offered two different approaches, because to my mind the term "théâtre à machines" is ambiguous in this context. Although the consensus here seems to be that it refers to a type of building, and it certainly could, it could perfectly well refer to a type of theatrical production. "Théâtre à machines" can mean either, but it much more often means the latter, as an Internet search will readily confirm: a theatrical genre, as for example in the following title of a major work on the subject:

Hélène Visentin, Le théâtre à machines en France à l'âge classique: histoire et poétique d'un genre (1999)
https://books.google.es/books/about/Le_théâtre_à_machines_en...

Though, as I say, it does sometimes mean a theatre building, as in the following example, from the Revue encyclopédique, ou Analyse raisonnée, vol. 12 (1821):

"Il entre dans beaucoup de détails sur la distribution et sur la construction des diverses parties d'un théâtre à machines"
https://books.google.es/books?id=WQcwAAAAYAAJ&pg=PP56&lpg=PP...

The ST, "un castelet reproduisant le théâtre à machines de l\'époque" could mean that the puppet booth (castelet) constructed by Řezač is a reproduction of a Baroque machine theatre building, but it could perfectly well mean that Baroque machine plays are reproduced in it. Personally I think it's more likely to mean the latter, partly because (as I've said) "théâtre à machines" usually means a type of theatrical entertainment, and partly because of the definite article, which suggests the genre to me; I think it would have been more natural to say "un théâtre à machines" (or "de machines") if it referred to a theatre. But of course this is not a decisive argument.

At any rate, if it does refer to the theatrical genre, then it's easy: "the machine theatre of the period" will do fine, or you could say "a machine play of the period" or "machine plays of the period".

"Any presentation which relies heavily on machinery can be referred to as machine theatre".
Martin Harrison, The Language of Theatre
https://books.google.es/books?id=ofbGhtZyiHwC&pg=PA147&dq="m...

"Circé, ou le Balet comique de la Royne (1581) foreshadowed many of the identifying features of this multigeneric machiine theatre"
John S. Powell, Music and Theatre in France, 1600-1680 (Oxford)
https://books.google.es/books?id=cm4NKuoIaBkC&pg=PA162

On the other hand, if we think it refers to a type of theatre building, it's not so easy to come up with a suitable term. The fact is that there is no term in common use in English for this type of theatre building that refers to machinery. Thinking about how I used to refer to them in the days when I taught and wrote about this subject, I think there's no doubt that the choice would have been "court theatre". To me, this is the functional equivalent. In the period we're talking about, a court theatre was inherently a theatre equipped with stage machinery and designed to put on machine plays (and operas, ballets, etc.). That's not to say that stage machinery was not also used in the public theatres, at this time and indeed earlier, but they cannot be called "théâtres à machines", and I think that if you say "reproducing a court theatre of the period" you are conveying what the ST is referring to (if indeed it's referring to a building). The sentence goes on to refer to the scenic resources: "doté de décors...", and I presume that what follows this clarifies what it's referring to.

See this account of Baroque theatre in Britannica for the customary terms for types of theatre building:
https://www.britannica.com/art/theater-building/Baroque-thea...

Of course you could expand the phrase to something like "a theatre equipped with stage machinery", but the sentence will be rather clunky if you do, and I don't think there's any need.