Se douter means indeed to suspect , but also, depending on the context, “to infer”, “to guess, based on the information one has”. This the meaning here.

Vous vous doutez que (and variants) is a common way to express politely what you expect your interlocutor thinks (or should think), something like “as you can guess…”

Ex:

Vous vous doutez que je ne travaille pas le dimanche. (= I guess you understand I don’t work on Sundays)

The translation “I need hardly tell you that I accept your suggestion most gladly” conveys that meaning with the same level of language. Maybe a more literal translation would be this: “you will certainly guess I accept your suggestion”.

“Se douter” means “to have some slight knowledge (of sth)” (“avoir une idée” in the definition below).

(TLFi) II. Emploi pronom. à valeur subjective. Se douter (de), (que). Avoir (une) idée de quelque chose, croire sur certains indices à une chose qu’on peut redouter. Être loin de se douter, avoir l’air de se douter de/que. (Quasi-) synon. conjecturer, deviner, pressentir, soupçonner.

Therefore “to have an inkling” corresponds well enough (OALD).

a slight knowledge of something that is happening or about to happen

• It is obvious that you must have an inkling that I accept your suggestion gladly.