The first thing to say is that I don't think you would have any success on the basis that the bus lane is badly signed and marked. There is the broad white line on the carriageway delineating it, and this is also painted on top of a broad red line. The sign in your photos is also visible, not totally obscured. Did you not notice the broad white line when driving along ? If this was the only basis for an appeal, you'd lose, in my opinion.
However, like so many dozy councils we have seen over the years, I think an appeal would succeed on the basis of fatal errors in the PCN. A PCN under the Transport Management Act 2004, and associated regulations named at the top of the PCN must contain certain information, but this one doesn't. We rarely see Bury bus lane PCNs, in fact this may be the first. It looks like they have updated the heading to show the latest regulations but have failed to actually read what those regulations have in them. Bus lane contraventions changed from separate legislation and regulations in May 2022, to come under the Transport Management Act.
I haven't looked in total detail at the PCN, but it omits one of the statutory grounds for an appeal, (Regulation 5(4)(f). The PCN must contain the statutory grounds,, so omitting this statutory content is a procedural impropriety and hence grounds for cancellation.
Here are the applicable regulations: -
https://www.legislation.gov.uk/ukdsi/2022/9780348232752/contentsand
https://www.legislation.gov.uk/uksi/2022/71/contents/madeCouncils never seem to think that the law applies to them as well as the recipient of the PCN; it's all just a nice little earner for them.
Of course, if you do decide to take them all the way to the Traffic Penalty Tribunal, the full PCN penalty will be in play, the discount option does not apply there.