You don't say whether the bike is 9 speed or 10 speed. If 10 speed, there aren't any significantly bigger cassettes that aren't horribly expensive (i.e., IRD). In that case, going compact is the better option. IMO this will look less horrendous anyway and gaps between the gears will be smaller. There are plenty of cheap cranksets on ebay and elsewhere...
If 9 speed you could use a MTB cassette and (probably) new chain. Sometimes a Shimano short cage road derailleur will work okay with a 32T rear cog, but usually not 34T. If you try this, all the gears may not work, and success will depend on the chain length, difference between chainwheels, dropout geometry, phase of the moon, etc.