Some sentences begin with a construction like "May I suggest that...." These are phrased as questions but are really statements. Should such a sentence end with a question mark or a period [full stop] (and what is the rationale)?
Sentences such as the one in your example may be grammatically interrogative, yet not be questions to which speakers expect the usual kinds of answer. There is a difference between their form (interrogative) and their function (statement or command). A question mark will be consistent with their grammatically interrogative form.
I guess this may be a question of personal choice and preference, but I at least would pronounce such constructs as questions, and would therefore punctuate them as such. To take examples similar to those in Edwin Ashworth's answer, I would end both Will you please bring me the papers? and May I suggest that Madame try the soup? with an upward inflection at the end. They are both requests, the first for a service and the latter for permission and they are both questions. At any rate, I would pronounce them as such.
This brings to mind that wonderful courtroom exchange (I've never been able to determine its veracity but it is very funny) between an incompetent barrister and a man accused of stealing 40000 hotel coat hangers:
Counsel: Yes, m'lud. Now, Mr Chrysler, perhaps you will describe what reason you had to steal 40,000 coat hangers?
Chrysler: Is that a question?
Chrysler: It doesn't sound like one. It sounds like a proposition which doesn't believe in itself. You know – "Perhaps I will describe the reason I had to steal 40,000 coat hangers... Perhaps I won't... Perhaps I'll sing a little song instead..."
Judge: In fairness to Mr Lovelace, Mr Chrysler, I should remind you that barristers have an innate reluctance to frame a question as a question. Where you and I would say, "Where were you on Tuesday?", they are more likely to say, "Perhaps you could now inform the court of your precise whereabouts on the day after that Monday?". It isn't, strictly, a question, and it is not graceful English but you must pretend that it is a question and then answer it, otherwise we will be here for ever. Do you understand?
Chrysler: Yes, m'lud.
Judge: Carry on, Mr Lovelace.
Note that this is even more ambiguous than May I suggest you try the soup? and is still punctuated as a question despite not being treated as one.
So, if you pronounce it as a question, punctuate accordingly. Janus Bahs Jacquet gave a very good example in his comment:
May I suggest, @terdon, that your example is perhaps not as ambiguous as certain other cases ‘in the wild’ are likely to be, and is therefore not really a very good sentence to use to exemplify this kind of usage?
It is long and convoluted but starts with a request for permission to make a suggestion, is therefore a question, and if I were to read it out loud I would pronounce it as such. Hence, the question mark.
I wouldn't use a question mark with May I suggest that you try the soup - I'd label this a hedged recommendation rather than a true question, despite the format. One certainly wouldn't be pleased with the obvious answers 'yes' or 'no'. Possible responses would be 'I'd prefer the prawn cocktail' or 'What is it?'
At 'The Question Is What Happened to the Question Mark? - Proof That ...' there is support for this position:
A polite request [/recommendation] will not use a question mark but will use a period instead.
Will you please get the attorney’s signature on this pleading and return it to me. (Not really a question because you expect them to do it.)
May I suggest that you research flight times before you book the travel.