The Arlington County Board begrudgingly offered a compromise settlement of its school-spending feud yesterday, but only after a Republican board member sharply attacked Arlington teachers as unprofessional and too political.

In a joint session with the school board, the County Board tentatively agreed to spend $45 million on the schools in the coming fiscal year. That would boost Arlington's share of school spending by 11 percent and give its 980 full-time teachers a 7 percent pay increase.

The County Board's two Democratic members dissented, with John G. Milliken accusing the three Republicans on the board of "trying to balance the [county] budget on the backs of schoolteachers."

"This is on the backs of taxpayers," countered Republican Dorothy T. Grotos, charging some teachers with using "children, little 5-year-olds" as part of a letter-writing campaign for increased school spending.

"Talk about teachers being professional! I'm not seeing much of that," Grotos continued. "They want to be political, political . . . Some of them certainly used students and parents for political purposes, and I think that is a disgrace."

School board member Torill B. Floyd defended the teachers, calling them "the most important civil servants we have. They should be given the greatest amount of respect . . . They shouldn't have to sustain derogatory remarks because they have to fight to keep up with inflation."

The 3-to-2 vote for the $45 million appropriation came along partisan lines and may be followed by a formal vote at the County Board's Saturday meeting. Under Virginia law, the school budget must be set by May 1.

Milliken, the County Board's junior member, said he will continue to press for a higher level of school spending, but yesterday's vote indicates he has a difficult task.

The compromise spending proposal, offered by County Board Chairman Stephen H. Detwiler, is $1.9 million less than the school board requested, but $1.5 million more than the $43.5 million the County Board had said it would be willing to contribute to the schools' $60 million budget.

Detwiler, a Republican, surprised the school board with the $45 million figure, saying it should be approved "in order to avoid further disruptions of school programs and finances."

After recommending the 7 percent salary increase for school employes, Detwiler also suggested that county employes be given a 9 percent pay increase instead of the previously proposed 12.1 percent -- a difference that Democratic board members Milliken and Ellen M. Bozman said would result in a $1.5 million savings.

The school board had recommended a 9 percent salary increase for school employes and a larger school board contribution to their health insurance premiums that would hve been tantamount to another 1 percent pay increase.

But Dewiler and his GOP colleagues Grotos and Walter L. Frankland said they were concerned about maintaining parity between the salaries of county and school employes. Last year school employes got a 10.5 percent pay increase while county employes got 8.5 percent.

Detwiler's suggestion drew moans from teachers wearing "Save Our Schools" buttons who had picketed outside the courthouse and then packed the meeting room. A few brought students from their civics classes to the meeting.

Several said they were concerned that the county Board would follow through on its threats to cut $3.4 million from the school board's package, an action they said would lead to only a 6 percent pay increase as well as higher pupil-teacher ratios and some teacher layoffs.

Detwiler said his proposal includes enough funds to maintain the current pupil-teacher ratio and to reduce it in grade schools.

In addition to the $45 million appropriation, the County Board indicated yesterday it will set aside between $900,000 and $1.2 million in contingency accounts for the schools to offset inflation and possible cuts in federal school aid proposed by the Reagan administration.