There does not seem to be any direct benefit, as the withdrawal agreement created a kind of six-month grace period after the end of the transition period (with some additional rules that are not really relevant to your case). The British government official web pages confirm that
The rights and status of EU, EEA and Swiss citizens living in the UK will ...
The benefit would be that you would be legaly residing in the United Kingdom after the 31st of December 2020.
With the exception of citizens of the Republic of Ireland, the long term residence status of EU Citizen in the United Kingdom will change when the transition period ends after the 31th of December 2020
Article 10 (1)(a)
The EU Settlement Scheme is ...
But my husband needs to register, is that right?
That is correct.
If he needs to register, will we need to show some rent agreement?
Possibly. See Carte de séjour de membre de la famille d'un citoyen de l'Union/EEE/Suisse. As far as I can tell, though, temporary accommodations should be sufficient, including those obtained through AirBnb. Specifically, ...
The Federal Public Service Foreign Affairs offers advice on this matter.
They, however, doge the question about tax consequences, but refer you to a site that deals with that matter.
Registration at a Belgian consulate | Federal Public Service Foreign Affairs
How to register
Returning to Belgium or moving to another country