Free access to Venice museums for the disabled, but with parking

Since your parking venue in Fusina is Parcheggio custodito 24h, you probably have a simple solution, at least to solve the Blue Badge problem. Make a photocopy of your Blue Badge, and take the original and the photocopy to the car park attendant. Since the car park is staffed 24 hours per day, you can ask them to check your original and then annotate the copy with their name and a message to indicate that the original has been checked.

In case you don't speak Italian, I would suggest language such as "Ho bisogno di usare questo tesserino per accedere dei musei. Per favore puo mettere suo nome sulla copia e la lascio sul cruscotto?". On the photocopy write something like: "Originale visto da Pinco Palino" and date and time. Pinco Palino would be replaced by the parking attendant's name. Then leave the photo copy on the dashboard.

Proof of disability in Italy is a very precise requirement. For example for the Vatican

Free entry, without the need to wait in line, is granted to all disabled visitors with certification of invalidity of 74% or over

Edit: This appears to be purely Vatican city rule and not aligned with Italian law stated below.

Needless to say, not too many foreign disabled visitors have a card with their percentage disability readily to hand.

The lack of a European-wide method to indicate disability was raised in parliament in 2013 regarding a proposal originating in 2011:

There are 83 million EU citizens with some kind of disability. They are entitled to enjoy the right to freedom of movement, as recognised by Article 26 of the Charter of Fundamental Rights of the European Union, as much as non-disabled EU citizens. While in their home countries there are a number of benefits in place for people with disabilities, such as concessionary fares, which facilitate their mobility and inclusion in all areas of social, economic and cultural life. However, these benefits cannot be transferred when travelling to another EU Member State, in order to work or go to university or for other purposes. This is a serious obstacle to the full enjoyment of the right of freedom of movement.

Which received the following answer (abbreviated):

The growing interest of EU Member States has enabled the Commission to initiate a project working group where representatives of interested Member States and civil society are dealing with practical details of issuing and managing a European model disability card. This group is still in the early stages of its work but the expectation is that the card to be developed is likely to grant benefits in the areas of culture, leisure, sport, transport and tourism.

In 2015 the following a "position paper" was published with the results so far:

incluD-ed asks the new EC, the European Parliament and EU MS to establish an EU Mobility Card providing “mutual recognition of disability status and thereby facilitating free movement of persons with disabilities in the EU

i.e. not a great deal.

Italian law providing access for the disabled says:

Possono inoltre entrare gratuitamente nei luoghi espositivi alcune categorie di persone tra cui:
...
i cittadini dell 'Unione europea portatori di handicap e ad un loro familiare o ad altro accompagnatore che dimostri la propria appartenenza a servizi di assistenza socio-sanitaria (Decreto Ministeriale n. 239 del 20 aprile 2006);

Which states essentially that free access sould be provided by law to the disabled and their assistants who can show they are eligible.

http://handylex.org expands on the precise requirements saying

Non è precisato quale documentazione sia necessaria per dimostrare la propria invalidità, né - soprattutto - come si accerti l'appartenenza a servizi di assistenza socio-sanitaria.

i.e. that the documentation required is not specified in the legislation.

In the absence of any precise requirement and any EU standard, the following anecdotes are offered:

http://boards.cruisecritic.co.uk/showthread.php?p=50053153

I travel a lot and visit lots of museums in the EU and have never been refused disabled concessions ( including Rome). I take a copy of my Blue Badge and DWP letter

I suspect the author is visibly disabled though.

https://www.euansguide.com/reviews/italy/lazio/rome/the-colosseum/review-this-is-how-is-could-be-done-nicola1972/

We were asked for proof of disability but had none (blue badge back in England), but as my husband has a clearly visable physical disability we were waved in to the shorter queue.

Free access to the vaporetto limited to wheelchair users:

Just tried it today. They won't accept any disabilities unless you have a wheelchair. (I live near Venice and work there).

http://boards.cruisecritic.com/showthread.php?t=2348207

Note: the Parking Badge is what was suggested to us by the ticket booth at the Vatican Museum as being adequate proof.

Other people have suggested a doctor's note, but I'm dubious about the value of a written letter.

In summary I suggest you use the method indicated at the beginning of this post to obtain the original Blue Badge to take with you. If you have any other official documentation, then obviously take that too.

This document may also be useful and provides links to the various Disability Benefits and Entitlements in European Countries

Buona fortuna!