it is gaming the companies system, because you are choosing to take sick leave as it is paid rather than emergency/compassionate leave which is not paid.
The company is stupid to have set their system up in this way - because it encourages people to take sick leave instead. In terms of employment law and employees rights there is no more requirement on a company to pay six months of sick leave at full pay than compassionate leave at full pay, they just happen to have written their contract in this way.
As a company, I would rather pay people to take parental or compassionate leave if it helps them equally. Sick leave has ramifications for planning (if its for a mental health issue) and for contact between company and employee, whether they can come into the office if they want to do that for any reason, etc.
It also means the OP has the mental load of repeatedly speaking to the doctor, making an appointment (takes over an hour on the phone at my surgery) and dealing with paperwork which is all entirely irrelevant when it basically boils down to her needing time off, rather than active medical care for herself (at this point). Also at six months, should her child still be ill, she will hit a cliff edge where all pay stops as her company offers six months sickpay. (perhaps seven months depending on when the sickpay period resets since we're in March)
If she takes something other than sick pay now however, even if its only agreed for a period, then if she is still equally stressed at the end of that agreement then that's where sick pay starts, doesn't it?
I really can't see any downside for the OP in an honest conversation with her employer (aside from a later update said she has and they were unhelpful) and many potential upsides of doing so - but you crack on and push her to discount any other options.
edit because heaven forbid I used a legally incorrect phrase