As a Bangladeshi passport holder in Ireland, I contacted the swiss embassy here in Dublin; they say I need to go to their embassy in London to apply for a visa, which is a bit weird for me because I need a visa to go to the even UK. If I apply for a Schengen visa to the Italian/German embassy here and visit Italy/Germany for a few days from Dublin, then with that same visa, can I travel to Zurich for a few days?
If you make your other visit(s) longer or more important than your visit to Switzerland, yes you can combine several Schengen countries into one trip and get the visa from an Ireland based embassy.
But remember that embassies do not like visa shopping (getting a visa from an other embassy than the one you actually should go to.)
If you go this route your other stay really has to be the longer one, as more important is hard to prove.
Can I advise that before you do anything further you contact the Swiss Embassy in London and ask directly if there is a specific process you can follow as a resident in Ireland? Do not contact TLS first as they are hopeless with these requests (for anything that is not straight forwards you won't be able to trust the answer they give you, even if they seem confident). It is possible that there will be a specific process to follow as a non UK resident. You would normally need to leave your passport for the processing which would also prevent you from leaving the UK so this leads me to think they may have a solution.
I speak from personal experience that the Embassy has a different process for the application of long term D visas for Irish residents than the websites or TLS are aware of and so I suspect they may also have similar for Schengen visas. It may not completely stop you from attending for biometrics but you may find that you only need to travel the once, or perhaps even by post the application if your biometrics are stored already from previous applications.
This is definitely the next step you should be taking as they may surprise you. The embassy is very busy, and may take a while to reply to emails, but you can call them.
Inform the Italian/German consulate, where you make the application, that you cannot make the application at the swiss consulate (your main destination), because they only offer such applications in an area (without an offer to a local alternative consulate) for applicants that require an extra, expensive visa, solely to apply for a Schengen Visa.
Your your travel plan should show that the consulate where you apply would be considered a 'second' best main destination.
As unfortunate as it is that 3rd country nationals, who are residents of the Common Travel Area (CTA), may not travel freely within this area may be - this should not lead into (effectualy) preventing an applicant to apply for an Schengen Visa without incuring undue costs (for a standard visa presently £100).