Do not let that $1000 goal force you onto a flight with two stops or a long overall flight time ( more than 20hours). These factors can seriously affect your jet lag experience..
We fly from Seattle to Europe every year and much prefer British Airways. Right now their price for a multi-city flight, into Venice and out of Rome, is $1065, only slightly over your goal, for a good flight combination with a reasonable overall flight time (around 15 hours). But note: almost all the BA flights from Seattle to Venice have you change airports in London, from Heathrow to Gatwick. Do NOT do this. There is one flight, leaving Seattle on BA 52 which departs around 1:20 pm, which gets to Heathrow in time (with a 2-hour connection) to make the morning flight from LHR to VCE. This is the only BA flight I would choose.
Coming back from Rome there should be no issue with changing airports, but make sure your London connection is through the same airport.
If you want to use your Alaska companion fare to reduce the overall cost, it adds some complications. You can fly from New York area airports into Venice and out of Rome on BA for $535 on your dates. Using your Alaska companion certificate to get to New York should cost around $500-600 for two, (I did not check your dates but that is what we generally pay). So the total per person should be around $850. But the risk of flying on separate tickets would not be worth the $200 savings to me. You are adding an extra stop, considerably extending the overall flight time, and subjecting yourselves to the risk of missing your long haul flight. Plus the certificate can only be used on planes operated by Alaska, and their flight schedule between Seattle and New York is limited. You would have to fly in the day before, incurring Hotel,expense, or take a very early morning flight from SEA and spend hours in the New York airport.
So I would advise you to save your companion certificate for other purposes. We only use ours if we want to spend several days on the east coast before flying abroad, such as last March when we went to Key West for a few days before flying out of Miami airport.
Another option from Seattle in the low-cost carrier class is Condor Air. They will fly you directly from Seattle to Frankfurt, where you will transfer to a Lufthansa flight to get to Italy, on the same ticket. They get mixed reviews and we have not flown them yet, but the fare is generally quite low.