By instinct, I would not use ate in that sentence.
British English prefers the structure with should + infinitive after suggest in the past. Cambridge does not even record the use of past tense after suggested:
Suggest + that-clause When we suggest an action to someone, we can use
a that-clause. In informal situations, that is often left out:
- I suggest (that) you leave here around four o’clock. The traffic gets
very bad from about 4.30 onwards.
When suggest is in the past, we can use should in the that-clause:
- Her doctor suggested that she should reduce her working hours and take
However, the use of past tense after suggested is not banned. On Englishgrammar you will find a description of how the verb suggest behaves. Among other things it says:
When the verb in the main clause is in the present tense, we can use a present tense in the that-clause after suggest. When the verb in the main clause is in the past tense, a past tense is possible in the that-clause as well.
- His doctor suggests that he stops smoking.
- His doctor suggested that he stopped smoking.
In both these examples that can be omitted.
At the question if the sentence
The boss suggested we left early yesterday.
is correct, BBC answers:
It is perfectly correct. You could also say:
- The boss suggested (that) we leave early.
- The boss suggested (that) we should leave early.
All these forms are possible with little or no difference in nuance, although the subjunctive form, suggested we leave, is a touch more formal.
There is yet another site that states that past simple is correct in British English after suggested:
In British English, we may use both the present simple and past simple
- Mark suggested that I called him.
All that being said, I think your sentence may well be called a covert mandative, but it can also be reported speech at the same time. Gngram finds instances of past after suggested, so it does exist. If the mother said
I suggest you just eat a banana.
the reported speech can occur as
My mum suggested I just ate a banana.
although according to most grammars the only options of reported speech are considered the following
- My mum suggested I just eat a banana.
- My mum suggested I should just eat a banana.