When there is evidence of DV (criminal conviction, police caution, letter from social services or a doctor etc) and you are on a low income you may be entitled to legal aid. Unfortunately if you aren't eligible for legal aid and can't afford a solicitor you will need to represent yourself.
If the father has Parental Responsibility both of you have the equal responsibility and rights to carry out the responsibilities and there is nothing you can do about that. Parental Responsibility is rarely revoked. It's unlawful to change your son's name without consent from all those with Parental Responsibility or permission from the court.
Allegations of DV don't carry much weight unless they are backed up with independent professional evidence e.g. police, social services, medical workers. DV isn't necessarily a bar to contact. The considerations are the effects of the DV on children, whether the risk of harm from DV or witnessing DV outweighs the harm children suffer when they don't have a relationship with one of their natural parents and what measures might be put in place to ensure that parenting is good enough. So your solicitor could be right and the father may well be granted contact.
When daytime care has been established it's difficult to argue against overnights on safety grounds. After all if the child is safe during the day why would he not be safe overnight?
You may have valid points about the father not knowing the routine or what to do if he hasn't been involved in caring for your son overnight but it's a case of thinking about what measures might be put in place so that he can learn to bring his parenting up to a "good enough" standard.
South Africa is a signatory to the Hague Convention so if the father abducted your son the child would be returned to the UK where the child habitually lives for the courts here to decide the way forward. If there is an immediate risk of international child abduction you can apply to court for a prohibited steps order to prevent the child being removed from the UK and for the passports to be held by a solicitor.
Moving away isn't a good idea. It could only make matters far worse and won't prevent contact or overnight stays. The courts have powers to order someone disclosures the whereabouts of a child. That means friends and family or government departments such as the DWP can be ordered to reveal a child's whereabouts.