Just to clarify a few points.
If she has been offered a caution and has a solicitor, she will already have been arrested, received legal advice, been interviewed and bailed.
She won't be paying privately as ALL police station advice is free. It is possible that she has paid for an extra appointment with the solicitor to go throug her options, but in terms of the solicitor being willing to 'fight' the case, it makes no odds.
Solicitors will generally advise someone to accept a caution where they admit the offence, in order to avoid going to court because the stakes are lower and a caution is spent after 5 years as opposed to 7 for a conviction. If someone desperately needs to avoid anything on their record, they might be advised that they can take the risk of going to court, on the understanding that they have very little chance of their gamble paying off.
Agood solicitor won't necessarily be able to get the case dropped. It will depend on the view taken by the individual prosecutor as to whether it is in the public interest to proceed. Some prosecutors are more bullish than others. Some are more reasonable than others. But I wouldn't expect any prosecutor to be in a hurry to drop a case involving DV, an admission in interview and CCTV.
The background between the two of them will have very little airing at a trial because most of it will be 'bad character'. Some may get into evidence simply because it's an intrinsic part of the case, but most would have to be the subject of an application. This almost certainly would fail because it doesn't go to the heart of theissue. If she was saying he was lying about her assaulting him, it might get in to demonstrate a history of false allegations etc. if she was asserting self defence, it might get in to demonstrate a history of physical aggression. But she doesn't actually have a defence in law, by the sound of things, and won't be allowed to bring in a whole lot of background in attempt to sway a bench against him.
It can be used in mitigation at sentence.
If she offers him money she could be arrested for perverting the course of justice, which is extremely serious.
To summarise, she can take a punt on going to court and having the case dropped, but I wouldn't fancy her chances. If she goes to trial on the basis of what has been said here, she will almost certainly be convicted, because she effectively admits the offence, and there is no defence of provocation in common assault.
Great sympathy for her, but she's walked right into his trap