The witness John Dowds being James' brother baptised in 1844 has to be a possibility and 'Handloom Weaver' is a perfect fit for the Gregory Dowds mentioned in that 1843 story in the Northern Whig. So we have Gregory and Ann Jane Dowds having a son John baptised on 30 November 1844, then marrying a couple of months later on 1 February 1845, then having another son James baptised on 26 December 1846, then... nothing, that's where the Irish records seem to fall off the cliff edge.
Civil death registration commenced in Ireland in 1864 and I can't readily see a death registration to fit Gregory or Ann Jane between 1864 and 1877. So a possibility has to be that they died before 1864, indeed, given the cliff edge I mentioned, they might even have died as early as the late 1840s. The bad news is that burial record coverage in Ireland pre 1864 is not at all good and in many cases is non existent, I checked 'RootsIreland' and wasn't surprised to draw a blank.
Edited to add: Here's something further on James' brother to look into if you like. John Dowds died of cardiac disease at a reported age of 68 on 8 December 1915 at 167 Renfrew Road, Glasgow, former residence 2 Houston Place, Glasgow. He was recorded as an engineer's labourer, with the comment that he was first married to Helen Gallagher, then to Mary Ann McFall or McCartney. His parents were recorded as Gregory Dowds, Hand loom Weaver and Ann McConville, both deceased. The informant was his widow Mary Ann of 11 Anderson Street, Kinning Park. The 'ScotlandsPeople' website has this marriage record and the 1883 marriage record for him and Mary Ann McFall or McCartney (RD Kinning Park).