Assuming Dec,2019 is your Gratuity eligibility month resignation on Sep and serving 3 months till Dec,2019 to get gratuity is ok.
With regard to your 3 months notice, please check your appointment order. Normally all Companies will have some clause in favour of them. Once resignation is submitted it implies that employee is not willing to continue his services with the Company and relieving from duties part goes to the hands of employer. If your employer has the clause of termination of employment with one month notice they may choose that option. Without seeing your appointment order clause can't comment on your question.
From India, Madras
As per the Judicial Dicta set down so far in respect of the interpretation of minimum qualifying service under the Payment Gratuity Act,1972, if an employee completes 240 days continuous service in his fifth year of service in the same establishment, he is eligible and entitled to gratuity as if he has completed not less than 5 years of continuous service.
Normally the minimum qualifying service is determined by the length of service rendered by the employee from his date of entry and the actual date of his exit. By implication, therefore, whatever the notice period, if any, is actually served by the employee due for exit without any buy-out, should also be included in the 240 days of the fifth year of service.
Your doubt about the HR's preemptive attempt to deprive you of your gratuity by accepting your resignation forthwith and arranging an early relief indicates the organization culture as well as the bad HR practice adopted under the undue influence of the particular HR. If it so happens, the move of the management would amount to a counter offer and they have to pay you 3 month's notice salary. Otherwise they will have to pay you gratuity for 5 years which may work out to an amount approximately equivalent to 2.5 months salary comprising of the sum of basic and D.A only. I hope the concerned HR would be a good mathematician and as such probably he would not venture to do so.
HR people are not descending from the heavens to inculcate ethics or morals to Managements. Mostly they are compelled to act as the watch dogs of the interests of the managements and not the human resources. SO, you only have to be beware of such things!
From India, Salem