I sound like an ideal guest for you as I'm in my 40s with a Mohican so I see where you're coming from
My DH wore jeans and I wore a basic maternity dress to our wedding (admittedly a couple of decades ago!) so we are definitely not cookie cutter wedding people.
But. (Sorry
) You ask "Would people get it? Not that it matters but I don't what to end up with an "off" atmosphere if people don't get our extremely relaxed and jokey theme"
The fact you ask this question - you know you have doubts! I think some parts of a wedding should be absolutely just simply about what the couple want - the ceremony and honeymoon mostly. But if you're throwing a party to celebrate you need to think about everyone else enjoying it not just yourself. Apart from the possible safety aspect of your dog's guests (can't believe I typed that
)mixing with small children...some people are just not going to get it.
I always think if you want a very individual specific niche-y kind of wedding, just elope and do it the two of you then you know that everyone there is definitely enjoying it 
If you want to go the route you are going, I would make it more a separate party to celebrate, then people aren't so bamboozled. If you are having a more formal wedding which has a seating plan the two styles (for me) aren't compatible. People won't get the "extremely relaxed" thing when they're being told where to sit and a formal invite etc.
You want to have a fun individual celebration party - totally go for it. Relaxed and jokey works for that. Invites and seating plans - not so much.
Also "after bagging my partner to get the ball rolling" worries me slightly, tbh. If you're begging (? Maybe you did mean bagging
) is he totally up for all this?
PS Also I think some unconventional people (for want of a better word, I really mean eccentric
) don't really get weddings. To me the ceremony itself is akin to a will writing or something legal. The party afterwards if you want one is a party and the two things are separate. I love to see people happy and smiling and in love but I hate a lot of wedding stuff like favours and decorated chairs and shit.
Maybe I'm not an ideal guest at all 
We got married mostly for legal reasons back then and surprisingly I really enjoyed it and I like being married (as much as a feminist does
) but I think people can have too much focus on the wedding not the marriage. Sometimes they seem more work than they should and perhaps yours may feel like that if you're worrying at this stage?