The vast majority of self-catering flats are only available for Saturday-to-Saturday lets, so you'll probably have to pay for a full week even if you're only staying for 5 days.
I don't know whether my web searching skill (or luck) is any better than yours but, for what it's worth, this is what I've come up with. (All prices are for late April, all locations are reasonably central and all have parking unless stated as 'unknown')):
Weyview Holiday Houses (�260 per week):
http://www.weyviewholidayhouses.co.uk/
Central Seafront Apartments (Prices unknown)
http://www.centralseafrontapartments.co.uk/index.htm
Queensway Apartments (�190 - �330 per week, depending on size & location of apartment):
http://www.queenswayapartments.co.uk/
Weymouth Harbourside Holidays (Prices unknown. Parking at Townbridge House only):
http://www.weymouthharbourholidays.co.uk/
Weymouth Harbourside Holidays - same name, apparently different company! (Prices unknown. Parking at Randall's 'Net Loft' & Cove Row Houses):
http://www.harboursideweymouth.co.uk/accommodation.html
Also check out Old Harbour Holiday Flats (prices and parking info unknown), Trezise Holiday Home (prices and parking info unknown) and Whitesands Seafront Apartments (prices unknown), from here:
http://www.weymouth.gov.uk/visitors/accommodation/listde tails.asp?svid=71
If I've only