What are some ways you can influence the roadmap as a product marketer when working with a strong-willed and difficult to work with product manager?

This question becomes more about how you can influence your PM vs. the roadmap yourself. Without knowing your history or the nature of your relationship I would advise for you to focus on building trust with this person.

Start by understanding his or her definition of product marketing and expectations from you. They may not have one because they haven’t worked with Product Marketing before and you’ll need to start by educating them on what PMK actually is in the context of your company. Find out what they need help with and deliver on that, consistently. Research? Competitive intelligence? A growth plan? First get into the groove with this person and then you’ll be able to influence.

In the cases where a Product Manager has consistently challenged your work beyond regular constructive collaboration, I have found the following tactics useful:

• Realign on company/product goals and strategy. It's possible that there's a gap in understanding or alignment that explains the pushback.
• Share your work and point of view more widely (beyond the Product team). This can be tricky depending on the organization, but sometimes, getting having your work seen by others can inspire renewed consideration.
• Ask for direct, 1:1 feedback. Because sometimes, it's may not be related to you at all...

First of all, most well-heeled product managers tend to be strongly opinionated but open to reasoning and change. The PM-PMM relationship is one of the most critical relationships for product people, and the success of the relationship has a direct impact on your success. Here are some methods i commonly use.

 
Effective questioning: My fav go-to strategy when dealing with a strong-willed or challenging to work with PM (or anyone for that matter) is the "Idea of Influence with good questions." I adopted this strategy as it moves the conversation to "educate me" on your approach. A vital element of this approach is to make the questions nonthreatening. A common and related approach is the 5-why approach. I find this approach to be my most effective method.

Research: Another approach that I commonly leverage is proof-driven reframing. Most PMs are data-driven individuals. In this regard having strong information such as (below) is very effective.
- Quantitative sales and field data
- Qualitative customer feedback
- Compete information
- AR feedback
 

Ownership: Having clear areas of ownership in terms of product development and GTM strategy clearly set the stone on who has the final word(and ownership) on any specific area of contention.

This can be tough - I feel you! But keep in mind all good relationships take time and if you can't figure out a good way to work together it's going to be really hard to succesfully do your job and vice versa. Here are my thoughts. 

Establish an equal partnership 

Sometimes PMs like to think that you work for them. It's important to make it clear you don't and that rather you both serve the product. Do this by making it clear that you have your own unique goals for the product based on sound research and data.

At the same time don't constantly second guess your product team / PM. You have to trust them to carry out the majority of the product decisions even if you don't agree with them. If there is somehting mission critical that's going wrong bring it up, but otherwise pick your battles and don't sweat the small stuff. Being a PM is a tough job and if there isn't a lot of built up trust they can get defensive towards a PMM. 

Give more than you get: 

This can be hard because PMMs often support many products but look for needs that the product has, that you can specfiically fill that the PM can't and go out of your way to deliver something great. This might be TAM research, or a persona guide. It'll build trust wtih your PM. 

Use verbatims and data: 

Product teams are always looking for the closest source of truth when making decisions. Try not to share feedback on the roadmap that isn't informed directly by a customer, research, or data. 

Don't take it personallly: 

You may have a great idea for the roadmap - it's just the product team has to balance many buffet of infomration to make decisions. If they don't implment a change don't take it personally but also ask why they did what they did. 

I could go on for days on this but hope this helps!