you definitely won't get thrown out for dancing with your Princess anywhere in the parks - thats lovely and I'm sure you'd probably get a round of applause
. You may not have room inside the castle, but you probably would on the balcony (and you definitely would by the wishing well, so many people don't even know it exists).
I'm not sure if a carriage ride up Main St would work with your plans though - the only horse-drawn vehicle at DLP is a tram, the other Main St vehicles are old engines ie a car, a police wagon, a firetruck etc. Theres no "carriage" as such. These vehicles are usually around in the mornings, but usually not very frequent after around 11am.
There is a stationary Princess carriage outside the Auberge de Cendrillon restaurant (where you go for the Princess lunch) - that might be a possibility for your romantic dance or even proposal?
Sorry you're having trouble making this a special Disney proposal, as you've found out DLP doesn't do in-your-face shout-it-out celebrations. I'm sure it will still be a very special day for your girlfriend, as you're putting so much effort into it
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eta - I saw photos from a couple who got engaged at DLP once - he got down on one knee with the ring when they had just reached Minnie on her queued meet n greet - I thought that was fabulous, because they then had the official photographers photos, the CM took photos with the couples camera, and Minnie herself really got into the part, being all excited for them!