Spotty- the aquarium is quite good, and the ticket lasts all day so you can pop in and out and to the beach next door if it cheers up for a bit.
Beaulieu is good and the ticket for the cars lasts for a year, so you could go more than once if the weather is bad all weekend! And the monorail is nice.
DD enjoyed the historic docks at Portsmouth. Again the ticket lasts for a year and we didn't quite manage everything in a day.
If you go to the new forest then the Red Shoot is family friendly and has reasonable food. And you can walk from around there.
If the weather is a bit nicer then a trip to Brownsea Island is good.
Taking note of other decisions though, for future trips to SIL.
Emily- a chair suitable from birth just means it lies flat, so look for a decent cosy toes type thing. John Lewis do some nice universal ones for about £15- certainly cheaper than £80 for an official icandy one!
And before deciding on a car seat which goes right through, have a look at the car seat boards about rear facing seats for stage 1/2. Not for everyone (expensive and bulky) but if you decided to go for it then you wouldn't want to spend extra on a first car seat which is extends through that age.
Mind you, if you don't use a car often then they're probably not worth it. But there are other seats which go from 9 months to age 12 and might work out better value in the long run.
If you though push chairs were a mine field...