Yes, high percentages of methanol in chloroform in dichloromethane can dissolve silica gel. Using a straight chloroform/methanol mixture, I find that 20% methanol is usually okay, and I have pushed it to 25%. but I wouldn't go any higher than that and expect not to find silica gel in the product. As Babcock_Hall says, redissolving the product in acetonitrile, ethyl acetate, or even DMF if you have the vacuum pump to rotovap it back off, will allow you to filter out the silica gel if you absolutely have to go that high.
I don't know what the addition of water and acetic acid would do, but I would expect that it would make it easier to dissolve the silica gel.
Chloroform:Ethyl Acetate:Acetic Acid mixtures might allow you to get a more polar solvent system without dissolving silica gel - I've never had a problem with silica gel dissolving in any other solvent I've tried.