Firstly, you can either marry in an Anglican church whereby you will not need to apply for permission to marry to the UK Border Agency (UKBA). If you cannot marry in a church, then you need to apply for a certificate of approval to marry because your partner is an overstayer. There is no Home Office fee for this application. The UKBA is not concerned about whether he is here legally or not, they focus on whether your relationship is genuine. You will need evidence to prove that you have been living together, eg utility bills, phone bills, mortgage details etc. It is also good to provide photographs of the two of you. If permission to marry is granted, he can then apply to remain on the basis of the marriage. However, you would need to show what are the obstacles for him returning to Jamaica to apply for entry clearance. The application to remain on the basis of your marriage will be outside the immigration rules because he is an overstayer. If granted, he would get 3 years discretionary leave to remain (he will be here lawfully and can work and travel etc). You need 6 years of discretionary leave before he would be able to apply for settlement (permanent stay) . If the application is refused, there would not be a right of appeal, unless human rights grounds are also raised. However, the UKBA are tending to issue only refusals to vary leave and not immigration decisions which would give you a right to appeal. You would have to challenge them to issue an immigration decision if this happens. You can also raise human rights grounds under Article 8 of the European Convention on Human Rights. This would be based on the family life you have established in the UK eg. his relationship with you and his child. The private life argument would be the length of time he has been in the UK, and his employment. If his child is a British Citizen then you would have stronger grounds to be argued. If the child is not British then you can still argue Article 8 family life. You mentioned that the child's mother has regular contact, so this is a fact that would be relevant to his human rights application.
To conclude, the only way to remain would be to make an application based on your marriage and raise human rights grounds. However, although he does not want to return to Jamaica to apply for entry clearance, if he did and he satisfies all the requirements of the rules, ( the main ones being intention to leave together, maintenance and accommodation without public funds) he would be given 2 years leave to enter. Before the 2 years is due to expire, he would be able to apply for settlement. Therefore, to remain in the UK and make an application will take him longer to obtain settlement than if he applied for entry clearance. Also, the fact that he overstayed cannot be used against him as a mandatory refusal because he is applying to return in a settled capacity.
I hope this is of help to you.