While I completely understand the wanting to respect your adult children to meet her, meet the girls, etc - they are also adults, and if this is imperative to them, they could go to Cambodia to meet them, see the culture which would help them understand them more, etc.
In addition if the wedding party is "pretty much a wedding" its possible that they could eventually think that you are married then you would waste all that time for nothing and have to start over.
The Adjustment of Status process would be around $2200 for your (future) wife and roughly $1600 for each of the two children that you would have to do AFTER coming here on the K1 and AFTER getting married but BEFORE they receive their green cards. That's a huge expense that could be fixed by having your kids go to meet her there if it's that important to them.
I don't disparage your reasons for wanting a K1, but many people around the world do get married without their families meeting their significant others because that's really the only choice they have (my husband had an unapproved/expired K1 prior to our successful spouse visa so we didn't have a choice). Your family should be able to understand your decision - in addition, if you did a spouse visa then YOU had an urgent situation you needed to return to the US for, your chances of being able to expedite for a spouse/step-children or just directly filing through the consulate is far greater.
I have adult children too so I get that you would like them to meet her - but at the end, they are adults, they don't live with you, and this is ultimately your life decisions. The spouse visa is such a superior visa for so many reasons - just meeting your family and having a US wedding (which you could still do at a later time) seems like a weaker reason for wanting to do the K1 (in my opinion)