I would query the dates you have given in this post. If Patrick was born in 1813, he’s not going to have married in 1823, when he was 10. I assume one of those years is wrong.
Researching him will be challenging. St Patrick’s is RC so I assume he was of that denomination.
The population of Belfast expanded dramatically in the 1800s. In it’s early years it was largely Presbyterian (being dominated by Scots who had moved to Ireland in the 1600s). In 1782 there were just 365 Catholics in the town. The first RC church (St Mary’s) only opened in 1784. The population of Belfast in 1800 was about 20,000. In 1901 it was 349,000. The establishment of the shipyards, 50 linen mills and countless support industries led to a huge influx of workers from the surrounding countryside who were looking for better paid jobs in the city.
Patrick might have been born in Belfast but it’s equally possible that his parents were amongst the tens of thousands who were pouring into the city every year around that time. Very few RC churches in Ulster have records pre 1825, so if they came from an outlying parish, then there will be no records. St Patrick’s in Belfast is an exception in having records from 1798 but St Mary’s which pre-dates it has nothing before 1867. (They either weren’t kept or have been lost). Researching RC families in the early 1800s is therefore very difficult.
Many labourers and farmers in Ulster grew flax which they then wove into linen, usually during the winter months when there wasn’t much work required on farms. They used hand looms such as are still used in the Outer Hebrides to make Harris tweed. They could be dismantled when not required and for moving. So those are the folk on the flax growers list. Probably three quarters of the rural population at that time grew a bit of flax. It provided a bit of top up income.
A flax dealer strikes me as something different. Clearly no flax was grown in a city, and I think a dealer would be someone who sold the seeds the plants were grown from. Presumably importing them through the port of Belfast might have been part of the process, though you can grow them in Ireland too.
I looked in the 1843 street directory (compiled 1842) but don’t see Patrick listed.
https://www.lennonwylie.co.uk/alphanames1843L2Mc.htm
There is a list of flax merchants but no McMullen listed:
https://www.lennonwylie.co.uk/1843TRADESP.htm
Not sure I can suggest any lines of enquiry that are going to turn up much on the family’s life in Belfast.