Will academic certificates from an unrecognized school be acceptable for police recruitment?

No. If you have shown that you have studied in a recognised school the study certificate from the same school. Alternatively you can get a certificate from a unrecognised school upto 9th class. Then at least you should get a certificate for tenth class study from the school the name of which is shown in your tenth class certificate.
But I don't understand how the recognised school accepted showing its name for a student who have not studied in their school. This type of issues will land you in troubles in many issues.

You have already got a certificate for your class 10 from the recognised school so you present that as one of the certificates.

Other certificates or credentials from class 4 to 9, you present from the ones which is given to you from the unrecognised institute.

In case you do not have the certificates from class 4 to 9 then go and contact the unrecognised school and tell them your requirement so that they give you a certificate that you have studied in that school and you have passed the classes from 4 to 9 during the years such and such.

I think this will solve your problem.

You need not worry in this regard since you have obtained the certificate of Board where in the certificate also carries the name of a recognised shool and hence your purpose is solved.
Now there are two ways of obtaining a certificate from a recognised school where you have studied up to class four and nine. In this connection, you may meet the Principal of the school indicated in your certificate and explain your situation. May be he would oblige you by providing such a certificate and for which you have to some cash for such a request.
The other way would be to meet the the authority of the school and request them to provide such a certificate from a recognised school. Since the Headmaster/ Principal of the school is having wide contacts with the different schools. Hence your work will be done after the interaction with the Headmaster/ Principal of the school where you studied.