You could always put an EQ pedal like a Boss GE-7 in your signal chain and set it up to scoop the mids a bit. British voiced speakers have more mids, and American voiced speakers have less mids, generally speaking. Country players also like speakers that don't distort. Hence the mention of the JBL. JBLs and EVs are known for staying super clean even at high volume.
Speaker switching can get quite expensive. Hence my mention of the EQ pedal option.
I looked around for whatever info I could find on the Ibanez GX60. Seems like a pretty low cost amp, solid state, from the early 80s. My wild guess is that an American voiced speaker with a high wattage rating would probably make it sound cleaner at high volume levels. However, at low volume levels, such as in your bedroom, the stock speaker probably sounds fine and your best bet is to adjust the EQ using the amp EQ controls, your guitar controls, and possibly an external EQ pedal to try and get the clean tone you're seeking.
I suspect this amp was designed to produce "shredder" tones that were popular back in the day, but at a lower price point.