If there's no RCE and no father's name on the person's marriage or death certificate, and the person was known always by his/her mother's name, there's unlikely to be a court case for paternity.
There is a small chance that there might be something in the Poor Law records, but only if the mother applied to the Parochial Board for relief.
SP has three databases for baptisms. One is the Old Parish Registers (OPRs), which are the registers kept by the Church of Scotland. These cover the period to the end of 1854, after which statutory civil registration started. The legislation that introduced civil registration empowered the Registrar General to collect all the C of S OPRs for safe keeping, these being the main records of births, marriages and deaths before 1855.
Then there are the Roman Catholic baptisms and the baptisms of what SP calls 'Other Churches', which do include some baptisms later than 1854. If you have not found your person in any of those then you are going to be looking for a needle in a haystack.
The child evidently didn't know its father's name because the name isn't on the child's marriage certificate, so one has to assume that the mother never let on who the father of her child was. What, then, are the chances that the mother, if/when she had the child baptised, revealed the name of its father?
Let us assume that this child was baptised, and that the baptism was recorded, and that the baptism includes the father's name. To find the record, you would have to approach the church, because the churches were never called upon to surrender their post-1854 registers. So what you need is a complete list of all the churches (of any denomination) where the child could have been baptised, and then you would have to apply separately to each one to find out where the register is, and get an extract.
It wouldn't be too hard to get a list of all the churches in Glasgow in 1909, because they will all be listed in the Valuation Roll. But tracking them all down would be very time-consuming, not least because many of them will have closed down or been amalgamated with others since 1909.
Frankly, I think that's a very tall order for an extremely small chance of finding the name of the child's father.
Sorry to be a wet blanket but I would not like to send you off on all sorts of wild goose chases.
