Why am I not seeing improvement in my bike split time?
Looking at your times, your times on the bike leg are below the levels you are setting in the other legs. But given that you are barely training for the bike leg, your times are pretty good.
This site says

How long does it take to finish a Half Ironman Triathlon?
The answer, based on our analysis of more than 67,000 finishers in 40 Half Ironman triathlons: about 6 hours, on average.  Swim 1.2 miles in 45 minutes, Bike 56 miles in 3:00, and Run 13.1 miles in 2:15.

In comparison, your swim time is better, your bike time is the same, and you run time is much better.
Your bike legs are 3 hour for 90 km; an average of 30 kph. Since the effort required for a given speed varies as the square (or some say cube) of the speed, riding 10% faster will take 20% to 30% more effort and energy.
To deliver that, you need to train your aerobic capacity to deliver oxygen for your muscles, and you need to train the muscles you'll use for the bike leg to deliver at that level for 3+ hours. These capabilities will not be improved by riding once a month - they will barely be maintained if you ride once per week.
Everybody will have their recommendations, reasons, and ideas. From my experience training with athletes of various levels, including one full Ironman competitor, here is a fairly conservative plan. For you own plan, work backwards from the race date. Work out how, and how much, and how long to taper. Work out what level of training you should be doing at the peak. Plan variation to keep it interesting. Set goals to measure you performance. Build up slowly to avoid over training and injury. And talk to a lot of people about it. A specialized coach would be best.
In the off-season, you should be riding a minimum of once per week for a minimum of 2 hours. About 4 months out from the race, that should step up to twice per week, with one longer ride and one shorter. Gradually increase the longer ride to about 4 hours at a lower intensity than your race pace to build endurance, and on the short ride warm up thoroughly before riding at race pace then cool down thoroughly. Gradually increase the period at race pace.
Two months out from the race, increase to riding 3 times per week. Continue the casual endurance ride as before, but increase the race pace ride gradually until the fast section is 2 hours. Six weeks before the race and 4 weeks before the race, push the race pace training ride so that you do the full course at race pace. The day after every race pace ride, do a short recovery ride. Be aware that people have differing opinions on the topic of recovery rides.
Be very careful not to over-train. Listen to your body. If you need a break, then do it. If you are getting injuries or feel jaded, back off. Assuming all is going well, 3 weeks out from the race is when it's time to get serious. A week earlier, you did the full course. This week do it again at race pace. The endurance ride becomes the previous race pace ride, and the recovery ride is also longer - there is more recovery to do. And now taper - reduce the length of the rides over the next week by 20%, by another 20% in the next week, and another 20% in the race week. Again, people have differing advice on tapering, so do your research - certainly some would suggest that another week of tapering would be better.