Collins hat:

arrogate in British English

(ˈærəˌɡeɪt )

verb

1. (transitive)

to claim or appropriate for oneself presumptuously or without justification

2. (transitive)

to attribute or assign to another without justification

Well, I'm not from 'arrogate meself, but ...

LOL

"Zuschieben" dürfte es treffen, es ist jedenfalls eine der Bedeutungen, die der Muret-Sanders für arrogate anbietet. Die anderen – "zuschreiben, zusprechen" – sind hier wohl zu positiv.

arrogate, v.

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3. transitive. With to, for. To claim (something) without justification on another's behalf; to assign or attribute (something) unduly, inappropriately, or incorrectly.

1584  W. Allen True Def. Eng. Catholiques viii. 198  The next step vnto which is (doubtles) to say and beleeue, that a temporal King is aboue the Priest in causes ecclesiastical..and so arrogate the regiment of the Church to a Queene.

1605  T. Tymme tr. J. Du Chesne Pract. Chymicall & Hermeticall Physicke i. vi. sig. D4v  We deny that those inset and naturall qualities..are to be arrogated to hotte, moist, and drie.

1818  S. T. Coleridge tr. in Friend (new ed.) I. iv. 34  To Antiquity we arrogate many things, to ourselves nothing.

1863  H. Cox Inst. Eng. Govt. i. viii. 111  An attempt was made..to arrogate to the Crown the privilege of issuing writs during a prorogation.

1953  Western Polit. Q. 6 814  They arrogated for Russian autocracy the role of freeing ‘the suffering Slav brothers’ from a foreign political yoke.

1991  Oxf. Art Jrnl. 14 i. 95/2  Advocates..tend to misread the modern and arrogate its defining characteristics to their own period.

2013  W. B. Hallaq Impossible State (2014) ii. 35  Arrogating to the social order an agency that stands autonomous from the state.

[OED]

Looking at the definition and example sentences in the OED at sense 3, I wonder if, in fact, Bercow’s actually using it correctly (first, the behavior he ascribes, rightly enough, to Johnson doesn't really consist in claiming sth. to or for another, second, Bercow uses no preposition). Whatever the answer to that, his sense is clear. Still, I think "assign" would be the standard verb to use.