Lifting the cap is a necessary step, but like some others I think more needs to be done.
Subsidised child care is important. As for education, if you're going to reduce the pressure I think it needs huge investment. There's far too much disparity between schools in urban and rural areas, and that hugely affects how your higher education progresses. In turn that increases stress on parents, which many people want to avoid. So I would:
1. increase investment in teacher training;
2. increase teacher salaries (to attract better quality candidates), with no variation for rural and urban areas (to try to stop the brain drain to the cities);
3. guarantee a spending level for each child so the schools get a minimum level of funding;
4. have top-up spending for children from poor backgrounds;
5. abolish hukou penalties for families - ensure that migrant workers can bring their children with them and get a good education in the cities. These are people the Chinese government should be rewarding, not punishing;
6. abolish tuition fees for public-sector schools or have the State pay for parents below a fairly generous wage;
7. abolish reserved places for local students at top universities - people born in Beijing do not need to go there.
If you do all this you will set the groundwork for an improved education system that people will feel more confident about, and in turn they'll think having a child isn't so bad.
As for a PR campaign, I doubt that would do much. For the last several decades the CCP has been pushing the story "growth, growth, growth - jobs, jobs, jobs - money, money, money". You can't change that with some positive stories about having children. Chinese people have to work hard even if they don't have children, so they can't be tricked into thinking having babies will be a breeze. Equally women aren't going to lie back and think of China.
If you really want to do something about perceptions, encouraging men to spend more time with child rearing and house work would be more productive, I think.
I think that would be both unfair and counter-productive. Not everyone is able to have large families, and it could encourage people to have lots of children they couldn't afford in expectation of getting a good job to pay for them. Then there's the fact you could end up with very incompetent people getting well paid, which would lease to worse outcomes for the company and resentment amongst other workers.
Perhaps what you meant or could have suggested was increased subsidies for parents. So, for example, their employers could offer free or subsidied child-care at their offices/nearby. This could be done privately or with financial assistance from the state. That way the parents find it easier to support their children and work but they're not getting ahead career-wise.
EDIT: Added in further reply below.
Not sure what stick you're thinking of, but punishing people for not having children if they have fertility problems would be immoral. Also I think that if you started restricting birth-control you'd end up with more women dying in back street clinics - which has happened in every country where abortions and the pill were banned or limited.
China has historically been used to the cultural preference for large families, and even today doesn't really have a lot of State-spending on children. As noted above it even punishes families if they're migrant workers. So I think it should try throwing money at the problem first. After all it's had a huge construction and infrastructure boom, where in many cases money was no object. Why not try that on people?