OP, I work in immigration law. As you probably know from your solicitor, even if you met the financial requirement (income of £18.6k per annum, British children do not count towards the financial requirement) you would not be able to apply for leave to remain in country as my understanding is you are here as a visitor. However, it is possible to make an application if you can rely on EX.1 of Appendix FM of the Immigration Rules:
"EX.1. This paragraph applies if
(a) [does not apply to you as you are in a relationship with the relevant children's father]
(b) the applicant has a genuine and subsisting relationship with a partner who is in the UK and is a British Citizen, settled in the UK or in the UK with refugee leave or humanitarian protection, and there are insurmountable obstacles to family life with that partner continuing outside the UK."
Rules are here:www.gov.uk/guidance/immigration-rules/immigration-rules-appendix-fm-family-members
If this paragraph applies, you are exempted from some of the normal requirements. It sounds like there are some grounds in your case, especially if your stepson has a relationship with his mother. Furthermore, it is always open to you to make an application outside the Immigration Rules relying on human rights law. If your solicitor has not discussed this with you please raise this with them or get a second opinion.
Petitions are not a common thing to recommend though, I am surprised that your solicitor suggested that to you. I doubt it will help, EX.1 applications are notoriously difficult. Prospects of success on appeal are very slightly better. You must ensure your solicitor completes your application properly and knows about the new appeal regulations to ensure you get a right of appeal if you are refused.
Finally, you may have been deceptive in what you have said at the border. For family based applications this isn't fatal on its own, but your solicitor must address this carefully. I hope this has helped. Please ensure you have a good legal representative.
To other posters, the financial requirement here sounds simple, but it is actually quite difficult to satisfy and many relatively well off families struggle because they cannot produce the correct evidence.
OP is not the first to be in this situation and won't be the last. I am not convinced about the persuasive value of a petition as evidence but OP is of course welcome to submit this with an application. As well as making an application within the Rules, OP has the right to make one outside the Rules. The Home Secretary retains a residual discretion to grant leave outside the Rules. In addition, she has to exercise her functions in respect of immigration having regard to the safeguarding and promotion of the welfare of children in the UK. It does not matter how those children came to exist, the reality is they do exist and their welfare is a key consideration.