Design or mechanical lien for interior design firm who isn't being paid?

Hi there –  We took on a project for a hotel that was seemingly in a tight spot after an interior design firm was under-delivering on the brief. The project was remodeling and improving amenities at a hotel affected by the California wildfires. We entered the project at the point they had general plans approved, but were lacking any real specific definition within each space. So our role was part interior design and part interior architecture. We agreed on a budget via email. We built a contract based on a clear brief from their General Manager. The only other person involved is the owner, who is part of an investment firm. When we shared the contract, they reviewed and generally said looks good (and agreed via email on budget etc). He said he would share with a lawyer friend of his though after they were "burned on their last contract". He later let us know that it was going to take some time, and that we were good to start on the project in the meantime if we were comfortable to. We have done this a lot in the past. Not ideal, but we tend to act with trust, especially knowing how much they needed our help. Because they were on a VERY tight timeline, we got to work quickly and within 4-5 weeks had performed a significant portion of the work, and help various design reviews with them. Everything from a new ADA compliant room design, to the overall look and feel of all rooms, to the new lobby design and an all-new lobby bar with outdoor seating area. They "loved" everything we shared – things seemed to be going great. We also consulted on a variety of other things, such as a potential new shop, and some exterior elements. So, week after week both us and the general manager kept asking the owner for the contract. After all of the above work, he responded with suggested amends from his lawyer. The amends were not anything anyone would reasonably agree to. Placed an extraordinary amount of liability on our shoulders (beyond the responsibilities of an interior design team), demanded various increases in scope that were clearly not part of the initial brief or verbal agreement, and changed payment terms to much smaller increments and with timing that meant we would only really be paid the bulk of the work at the end. In reality, the bulk of the work was very much up front. We provided a very reasonable response, hoping perhaps he didn't understand what his lawyer had suggested. Unfortunately, we think his lawyer was an excuse for his own lack of trust etc. We got a very bad attitude when we tried to discuss. He failed to really respond to anything and basically said it is what it is. After thinking about it, we decided to let them know we are no longer in the position to complete the project, but would appreciate payment for our work to date. They had issued roughly half of the amount of our work done to date. Meaning they owed the other half of that. I broke it down for them, even discounted some of our efforts that went beyond expectation (because we're nice). We didn't hear back from them at all. We followed up after that, and still no word. It has now been 2 months since that time and we don't think they intend to pay. They are definitely in construction because they started demolition and other prep while we were actively designing, and are on a tight schedule. We would like to send a notice of intent to lien. We are unclear if this would be best as a design professional lien or mechanics lien? And if there are any other considerations here because they never effectively signed the contract? (But did clearly agree to things via email before we kicked off.) Thank you for your help, much appreciated.