There are two separate strands to the answer to "Anyone know where I stand legally?". One relates to the management company and the other to the tenant.
Firstly, while your contract with the management company probably allows them to employ contractors on your behalf (and then send you the bill), they must also act in your best interests when doing so. That means, for example, that they can't simply employ someone who charges a fortune for a very simple task; they need to seek the lowest quote.
You've stated that the local council wouldn't take the items away. That doesn't surprise me as most councils won't do it for free. However I've never heard of a council that doesn't offer a paid service for collecting bulky waste items. For example, Sheffield City Council would have charged £40.20 for collecting the bed and the chair:
https://www.sheffield.gov.uk/environment/waste/bincollections/bulkycollections.html
Assuming that the relevant council (where the flat is) has similar charges, you've been overcharged by £90.
So I recommend checking the local council's website to find out what they would have charged for the job. (If you can't find the information, tell us which council area it is and I'll see if I can find it). Then tell the management company that you'll only pay that amount, since you're not responsible for their failure to seek the lowest price.
With regard to the former tenant, he is responsible for recompensing you for any losses that you've suffered through his actions. So you can send him a written demand for compensation and, if necessary, take legal action against him to get your money. However, just like you, he can then say "I'm prepared to pay £40, because that should have been the cost of removing the items, but I'm not paying £130 just because you, or your agent, didn't bother to find the lowest price".