Is this possible and how do I go about it without another refusal?
An application is always possible, it's a matter of filling out the form and paying the fee and biometrics. Your question is more directed to whether or not a T2 refusal will make them worry when you apply for a Standard Visitor Visa (SVV). You tried to renew a T2 while the cooling-off period was active. Automatic refusal; it's a condition you agreed to at the outset and jeopardises your employer's standing with UKVI. What in the world were you thinking? Never mind, water over the dam...
I took a course in T2 a while back and this question came up (i.e., how does a T2 refusal affect other applications). The answer is indeterminate because there are so many factors to consider, mostly the premise. The instructor, however, offered a general rule of thumb...
A T2 refusal will generally invite further refusals for any
applications junior to T2.
'Junior' in this case means T4, T5, and SVV. This does not mean there is an automatic refusal or even that your own circumstances will not qualify. It's a general rule of thumb that will cause practitioners to treat the premise with extra diligence and not to undertake the casework if the premise is weak. They tend to get somewhat persnickety about red flags.
Your premise is two-fold: you want to (a) maintain a relationship and (b) clear your flat. You cannot expect them to be sympathetic to clearing out your flat because the ECO is going to think you would have had that opportunity already and didn't do it because you wanted to create a premise for returning to the UK. Or worse, that you contemplate living there in the future. If the ECO concludes that you have contrived a premise, you will not be back, even when your cooling-off period has expired. He may also ask an assistant to contact your landlord and find out if he has been compliant in the renting rules for T2 migrants. If your lease is still held and you are on an expired T2, he may be in trouble. You can ask for more information about this aspect on the Expats site (we don't know anything about T2 specifics).
Your other premise, i.e., to maintain a relationship, is generally fine, it's normal for people to build and maintain relationships. But given your history thus far the ECO is entitled to suspect that you have been building up a private life in the UK and were using a T2 for that purpose, especially given a premature application. If things go in that direction, they may send a T2 inspector around to your employer to discuss it further. Of course he may conclude that you were playing it straight-up, but it's always good to know what is going through an ECO's mind.
The answer to your question is indeterminate. Your narrative, however, indicates that elevated diligence (very elevated in fact) is needed for you to establish a credible premise. If in doubt, consult a practitioner or don't apply.